Vectors NEET Questions

Explanation: This question asks for the angle between a Vector lying in the XY-plane and the Z-axis direction. Since the given Vector has only i and j components, it has no component along the Z-axis, which is crucial for determining orientation.

Vectors are defined in three-dimensional space using components along x, y, and z axes. The angle between a Vector and an axis depends on the projection of the Vector along that axis. The direction cosines help in relating Vector components to angles with axes.

To find the angle with the Z-axis, we use the formula involving the dot product between the given Vector and a unit Vector along the Z-axis. The Z-axis unit Vector is k, and the dot product depends only on the k-component of the given Vector. Since there is no k-component present, the dot product becomes zero. When the dot product is zero, it implies the Vectors are perpendicular.

Think of a flat sheet lying on a table (XY-plane). The Z-axis is like a vertical stick rising upward from the table. Any vector drawn on the table surface makes a right angle with that vertical direction.

Thus, a vector confined to the XY-plane is perpendicular to the Z-axis, forming a right angle between them.

Explanation: This problem involves comparing two fundamental vector operations—dot product and cross product—and using their magnitudes to determine the angle between Vectors.

The dot product of two Vectors depends on the cosine of the angle between them, while the magnitude of the cross product depends on the sine of that angle. Specifically, the dot product is proportional to cosθ, and the cross product magnitude is proportional to sinθ.

Given that the magnitudes of these two quantities are equal, we equate the expressions involving sinθ and cosθ. This leads to a trigonometric condition where the sine and cosine of the angle have equal values. Such equality occurs at a specific angle in the first quadrant.

From trigonometry, when sinθ equals cosθ, dividing both sides gives tanθ = 1. The angle satisfying this condition is well-known and lies between 0° and 90°.

A simple way to visualize this is to imagine a right triangle where the opposite and adjacent sides are equal. This creates a specific angle where both sine and cosine ratios become identical.

Thus, the condition of equal dot and cross product magnitudes uniquely determines the angle between the Vectors.

Explanation: The problem requires finding a vector that has the same direction as a given vector but a different magnitude. Such vectors are called parallel vectors, meaning one is a scalar multiple of the other.

A vector can be scaled by multiplying it with a constant. This changes its magnitude but not its direction. First, the magnitude of the given vector is determined using the square root of the sum of squares of its components. Then, a unit vector in the same direction is obtained by dividing the vector by its magnitude.

Once the unit vector is found, it is multiplied by the desired magnitude (51 units). This gives a new vector that maintains the same direction as the original but has the required length.

Imagine an arrow pointing in a certain direction. Making it longer or shorter without changing its direction gives parallel vectors.

Thus, by scaling the unit vector appropriately, we obtain a vector parallel to the given one with the specified magnitude.

Explanation: This question focuses on identifying vectors that are parallel in three-dimensional space. Parallel vectors differ only by a scalar multiple and point in the same or exactly opposite direction.

If one vector is a scalar multiple of another, then their corresponding components must have the same ratio. To check parallelism, we compare the ratios of the coefficients of i, j, and k for the given vector and each option.

If all three component ratios are equal for any option, that vector is parallel. If even one component does not match the proportionality, the vectors are not parallel.

Think of parallel vectors like arrows pointing along the same line but possibly with different lengths or directions (same or opposite).

By checking proportionality of all components consistently, the vector that maintains this ratio with the original vector is identified as parallel.

Explanation: This problem involves finding a vector perpendicular to two given vectors. In three-dimensional vector algebra, such a vector is obtained using the cross product.

The cross product of two vectors produces a third vector that is perpendicular to both. Even if the original vectors lie in a plane, their cross product gives a direction normal to that plane.

After computing the cross product, the resulting vector may not be of unit length. To convert it into a unit vector, its magnitude is calculated and each component is divided by that magnitude.

Visualize two vectors lying flat on a sheet. The perpendicular vector would point directly upward or downward from that sheet.

Thus, by taking the cross product and normalizing it, we obtain a unit vector perpendicular to both given vectors.

Explanation: This question explores the directional property of the cross product. When two vectors are perpendicular, their cross product produces a vector perpendicular to the plane containing them.

The X-axis and Y-axis lie in the same plane, known as the XY-plane. The cross product of vectors along these axes will therefore be perpendicular to this plane.

The direction of the cross product is determined using the right-hand rule. Curling the fingers from the first vector toward the second gives the direction of the resulting vector along the thumb.

Imagine placing your right hand so that fingers point along the X-axis and curl toward the Y-axis. The thumb then points in the direction of the resulting vector.

Thus, the cross product of vectors along the X and Y axes results in a vector perpendicular to their plane.

Explanation: The cross product of two vectors in three dimensions is calculated using a determinant involving unit vectors i, j, and k. This operation results in a vector perpendicular to both original vectors.

To compute it, a matrix is formed with the unit vectors in the first row, components of A in the second row, and components of B in the third row. Expanding this determinant gives the resulting vector.

Each component of the result comes from calculating minors and applying appropriate signs. Care must be taken while handling signs, especially for the j-component which carries a negative sign.

Think of it like a structured expansion similar to solving determinants in algebra, but applied to vector components.

By systematically expanding the determinant, we obtain the final vector representing the cross product.

Explanation: Torque is defined as the cross product of the position vector and the force vector. It represents the rotational effect of a force about a point.

The position vector indicates where the force is applied relative to a reference point. The torque is calculated as S×F, using the cross product method.

To compute this, we again use the determinant method with unit vectors and components of S and F. The resulting vector gives both magnitude and direction of torque.

Imagine using a wrench to rotate a bolt—the farther from the pivot point and the more perpendicular the force, the greater the torque.

Thus, by taking the cross product of position and force vectors, the rotational influence of the force is determined.

Explanation: This question asks for a vector that has a specific magnitude and is directed opposite to a given vector. Opposite direction means the vector points exactly in reverse.

First, the magnitude of the given vector is calculated using the square root of the sum of squares of its components. Then, a unit vector in the same direction is found.

To reverse the direction, the unit vector is multiplied by -1. After that, it is scaled to the required magnitude by multiplying with 10.

Think of an arrow pointing in one direction—flipping it exactly backward gives the opposite direction, and stretching or shrinking changes its length.

Thus, by reversing direction and scaling appropriately, the required vector is obtained.

Explanation: This question examines a condition involving the sum and difference of two vectors being perpendicular. When two vectors are perpendicular, their dot product equals zero, which is the key idea here.

Let the vectors be A and B. The condition is (A + B) · (A − B) = 0. Expanding using distributive law gives A · A − B · B.

Since the dot product of a vector with itself equals the square of its magnitude, the expression becomes |A|² − |B|² = 0. This simplifies to |A|² = |B|², meaning both vectors have equal magnitudes.

You can visualize this as two forces of equal strength acting in different directions such that their combined geometric relationship satisfies the perpendicular condition.

Thus, the given condition leads to a specific relationship where both forces must have equal magnitudes.

Explanation: This problem focuses on angular momentum, which is defined as the cross product of the position (displacement) vector and the linear momentum vector. It represents the rotational effect of motion about a point.

The formula used is L = r × p, where r is displacement and p is momentum. To compute this, we use the determinant method involving unit vectors i, j, and k along with the components of both vectors.

Each component of the resulting vector is obtained by expanding the determinant carefully, ensuring correct signs, especially for the middle term. The final result gives a vector perpendicular to both displacement and momentum.

Imagine a particle moving in space—the way its position and momentum combine determines how it tends to rotate about a reference point.

Thus, by applying the cross product, the angular momentum vector describing rotational motion is obtained.

Explanation: This question asks for a vector that is perpendicular to a given vector in a plane. Two vectors are perpendicular if their dot product equals zero, which forms the basic condition for solving this problem.

For a vector of the form ai + bj, a perpendicular vector can be obtained by interchanging the coefficients and changing the sign of one component. This ensures that their dot product becomes zero.

To verify, we multiply corresponding components and add them. If the result is zero, the vectors are perpendicular. This is a direct application of the dot product property.

Think of directions like east and north—they are at right angles and do not interfere with each other’s direction.

Thus, by constructing a vector that satisfies the zero dot product condition, a perpendicular vector is identified.

Explanation: work done by forces in vector form is calculated using the dot product between force and displacement. When multiple forces act, their resultant force is considered.

First, the two given force vectors are added to obtain a single resultant force. Then, the dot product of this resultant force with the displacement vector is calculated.

The dot product involves multiplying corresponding components and summing them. This gives a scalar value representing the total work done.

Imagine pushing an object using two forces at once—the combined push determines how much work is done over a displacement.

Thus, by combining forces and applying the dot product with displacement, the total work done is determined.

Explanation: This question is based on the dot product definition of work. work is given by W = F · S, which depends on the magnitude of force, displacement, and the cosine of the angle between them.

If work is zero, then either the force or displacement is zero, or the cosine of the angle between them is zero. Since force and displacement are present, the condition reduces to cosθ = 0.

When cosθ = 0, the angle between force and displacement is 90°, meaning they are perpendicular. In such a case, no work is done because the force does not contribute in the direction of motion.

For example, carrying a load horizontally involves an upward force but horizontal displacement, resulting in zero work by that force.

Thus, zero work implies that force and displacement are perpendicular to each other.

Explanation: The area of a triangle formed by two vectors is half the magnitude of their cross product. This method is widely used in vector geometry.

First, the cross product of the two vectors is calculated. In two dimensions, this results in a vector perpendicular to the plane with only a k-component.

The magnitude of this cross product gives the area of the parallelogram formed by the vectors. Dividing this value by 2 gives the area of the triangle.

You can imagine two vectors forming adjacent sides of a triangle—the cross product helps determine the space enclosed by them.

Thus, by computing half the magnitude of the cross product, the area of the triangle is obtained.

Explanation: For a body to be in equilibrium, the vector sum of all forces acting on it must be zero. This is a fundamental condition in mechanics.

The given forces are added component-wise along i, j, and k directions. The sum of coefficients in each direction must equal zero.

This results in a SET of linear equations involving the unknown ‘a’. Solving these equations simultaneously gives the required value.

Think of it like balancing forces so that the object remains at rest—no NET force acts in any direction.

Thus, by applying equilibrium conditions and solving component equations, the value of the unknown constant is determined.

Explanation: This question tests the distinction between scalar and vector quantities. Scalars have only magnitude, while vectors have both magnitude and direction.

Physical quantities like energy, temperature, and Mass are scalars because they do not depend on direction. In contrast, quantities like velocity and force require direction for complete description.

To identify a scalar, we check whether the quantity can be fully described by a single numerical value without specifying direction.

For example, temperature is the same regardless of direction, whereas velocity changes with direction.

Thus, scalar quantities are those that are completely described by magnitude alone.

Explanation: Specific Heat capacity is defined as the amount of Heat required to raise the temperature of a unit Mass of a substance by one degree. It combines concepts of energy, Mass, and temperature.

The SI unit of Heat is joule, Mass is kilogram, and temperature is measured in kelvin. Combining these units appropriately gives the unit of specific Heat capacity.

This quantity tells how much energy is needed to change temperature, which varies for different materials.

For example, water has a high specific Heat capacity, meaning it requires more Heat to raise its temperature compared to many substances.

Thus, by combining units of energy, Mass, and temperature, the SI unit of specific Heat capacity is determined.

Explanation: This question involves unit conversion from microns to meters. A micron is a unit of length commonly used for very small distances, especially in Optics.

One micron is equal to 10⁻⁶ meters. To convert a value from microns to meters, it is multiplied by this conversion factor.

The given wavelength is expressed in microns, so multiplying it by 10⁻⁶ converts it into meters. Care must be taken to handle powers of ten correctly.

Think of it as converting millimeters to meters, but on a much smaller scale.

Thus, by applying the correct conversion factor, the wavelength is expressed in standard SI units.

Explanation: Errors in experiments are generally classified into systematic, random, and personal errors. This question focuses on identifying the type of error caused by fluctuations in voltage.

Systematic errors occur due to consistent and predictable factors such as faulty instruments or environmental conditions. Random errors arise due to unpredictable variations, while personal errors are due to human mistakes.

Voltage fluctuations are external environmental changes that affect measurements in an irregular manner. Since these variations are not constant and change unpredictably over time, they introduce uncertainty in readings.

For example, if a power supply keeps varying slightly, the readings of instruments connected to it will fluctuate randomly.

Thus, such unpredictable variations in voltage lead to errors that fall under the category of random errors.

Explanation: This question deals with the concept of significant figures and measurement accuracy. The number of significant figures indicates the precision of a measurement.

A measurement with more significant figures is considered more precise because it provides more detailed information. However, the last digit in any measurement is always uncertain, representing the limit of precision.

The assertion compares measurements with increasing decimal places, while the reason explains the uncertainty in the last digit. The relationship between these statements depends on understanding precision and uncertainty correctly.

For instance, writing 0.5000 m suggests a more precise measurement than 0.5 m, but the uncertainty still exists in the last digit.

Thus, analyzing both statements requires understanding how significant figures reflect measurement precision and uncertainty.

Explanation: This problem involves calculating the average absolute error from multiple measurements. It helps estimate the reliability of repeated observations.

First, the mean value of the given measurements is calculated. Then, the absolute deviation of each reading from the mean is found by taking the positive difference.

These deviations are summed and divided by the number of observations to obtain the average absolute error. This gives an idea of how much individual readings vary from the mean.

For example, if measurements are close to each other, the error will be small, indicating higher precision.

Thus, by calculating mean and deviations, the average absolute error is determined.

Explanation: This question involves multiplication of measured quantities and expressing the result using correct significant figures.

When multiplying values, the result should have the same number of significant figures as the quantity with the least number of significant digits. This ensures consistency in precision.

First, the area is calculated by multiplying length and breadth. Then, the result is rounded off based on the least precise measurement.

For example, if one value has three significant figures and another has four, the final answer should be expressed in three significant figures.

Thus, by applying multiplication rules and rounding appropriately, the correct area is obtained.

Explanation: This question is about identifying derived SI units. Joule is the unit of energy, and dividing energy by time gives a rate of energy transfer.

In Physics, the rate at which work is done or energy is transferred is called power. The SI unit of power is defined as one joule per second.

This relationship connects energy and time to form a derived unit. Recognizing such combinations is important for understanding physical quantities.

For instance, electrical appliances are rated in terms of power, which tells how quickly they consume energy.

Thus, joule per second corresponds to the SI unit of power.

Explanation: This problem involves error analysis in calculating acceleration due to gravity using a simple pendulum. The formula for g depends on length and time period.

Errors in measurement of length and time contribute to the overall error in g. The fractional error in g is calculated using the relation involving fractional errors in length and time.

Since time is measured over multiple oscillations, the error in time per Oscillation reduces. Both errors are expressed as ratios and combined appropriately.

For example, measuring time over many oscillations improves accuracy by reducing relative error.

Thus, by combining fractional errors of length and time, the overall accuracy in g is estimated.

Explanation: This question examines the suitability of units for measuring very small physical quantities like nuclear radii.

Nuclear dimensions are extremely small, typically of the order of 10⁻¹⁵ meters. The fermi is defined as 10⁻¹⁵ meters, making it convenient for such measurements.

Units like micron (10⁻⁶ m) and angstrom (10⁻¹⁰ m) are much larger in comparison and not suitable for expressing nuclear sizes precisely.

For example, using a very large unit to measure a tiny object would lead to inconvenient decimal values.

Thus, choosing an appropriate unit like fermi simplifies representation and improves clarity in nuclear Physics.

Explanation: Latent Heat represents the amount of heat required to change the phase of a substance without changing its temperature. It is essentially energy per unit Mass.

The dimensional formula of energy is M L² T⁻². Since latent heat is energy divided by Mass, its dimensions are obtained by dividing by M.

This gives L² T⁻² as the dimensional formula. To identify a similar quantity, we compare this with dimensions of other physical quantities.

For example, some energy-related quantities share similar dimensional forms when normalized by Mass.

Thus, by analyzing dimensions, latent heat can be compared with other quantities having the same dimensional expression.

Explanation: This question involves addition of measured quantities and expressing the result with correct significant figures.

In addition or subtraction, the result should be rounded off to the least number of decimal places among the given values. This ensures consistency in precision.

Here, one value has more decimal places than the other, so the final result must match the least precise measurement.

For example, if one value is measured up to one decimal place, the result should also be expressed up to one decimal place.

Thus, by applying rounding rules based on decimal places, the correct sum is expressed.

Explanation: This problem involves finding magnitudes of two forces given their ratio, angle between them, and resultant.

The formula for the resultant of two forces is R² = A² + B² + 2AB cosθ. Using the ratio, one force is expressed in terms of the other.

Substituting these values into the formula allows solving for the unknown magnitude. Once one force is found, the other follows from the ratio.

Think of two forces acting at an angle—resultant depends on both their magnitudes and the angle between them.

Thus, by applying the resultant formula and ratio, the magnitudes of both forces are determined.

Explanation: This problem involves relative velocity in river motion. The speed of a boat upstream is reduced due to the stream, while in still water it moves at its own speed.

Let the speed of the boat in still water be considered. The upstream speed becomes (boat speed − stream speed). The time taken upstream is distance divided by this reduced speed.

For the return journey in still water, the speed remains unchanged, so time is calculated directly using distance and boat speed. The total time is the sum of both intervals.

By forming an equation using total time and simplifying, the unknown speed is found. Then, the required time in the stream is calculated accordingly.

Thus, by applying concepts of relative velocity and time relations, the required duration is determined.

Explanation: This question is based on relative velocity. The apparent direction of rain changes due to the motion of the observer. The man must align the umbrella along the direction of the relative velocity of rain with respect to him.

The vertical component of rain represents its actual falling speed, while the horizontal component is due to the man's motion. The ratio of these components determines the angle of inclination.

Using trigonometry, the angle is found by taking the ratio of horizontal to vertical components. The given condition that rain appears twice the speed of the man helps establish this ratio.

Imagine running in rain—raindrops seem to hit your face at an angle instead of falling straight down.

Thus, the umbrella must be tilted in the direction of the resultant relative velocity to avoid getting wet.

Explanation: This question tests understanding of scalar quantities, which are defined as physical quantities having only magnitude and no direction.

Examples of scalar quantities include Mass, temperature, energy, and electric current. These can be completely described using a numerical value and unit alone.

In contrast, vector quantities like velocity, force, and acceleration require both magnitude and direction for complete description.

For instance, temperature at a point does not depend on direction, while velocity changes if direction changes.

Thus, a scalar quantity is one that is independent of direction and described fully by magnitude.

Explanation: Vector quantities are those that require both magnitude and direction for their complete description. This question focuses on identifying such a quantity.

Examples of vectors include displacement, velocity, force, and Electric Field. These quantities cannot be fully described without specifying direction.

Scalar quantities, on the other hand, like Mass or energy, depend only on magnitude. The distinction lies in whether direction plays a role.

For example, force applied in different directions produces different effects, making it a vector quantity.

Thus, a vector quantity must include both magnitude and direction in its definition.

Explanation: This question explores the geometric interpretation of vector addition in three dimensions. Vectors A and B lie in one plane, while C lies in a different plane.

The sum A + B lies in the plane of A and B. However, adding vector C introduces a component outside that plane. This means the resultant vector does not remain confined to a single plane.

In three-dimensional space, vectors from different planes generally combine to form a resultant that is not restricted to either plane.

Imagine combining directions from different surfaces—the final direction points somewhere in space, not just along one flat surface.

Thus, the resultant vector does not lie entirely in a single defined plane.

Explanation: Two vectors are perpendicular if their dot product is zero. This condition is used to determine the unknown constant in the vectors.

The dot product is calculated by multiplying corresponding components and adding them. Setting this equal to zero gives an equation involving the unknown.

Solving this equation provides the required value. Care must be taken to correctly handle coefficients of i, j, and k components.

For example, perpendicular directions like north and east have zero dot product.

Thus, by applying the perpendicular condition and solving the resulting equation, the unknown value is obtained.

Explanation: Physical quantities are often classified based on how they are represented mathematically. Scalars and vectors are common, but a more general representation exists.

Some quantities require more complex descriptions, involving multiple components and directions, especially in advanced Physics. These are represented using tensors.

Tensors generalize scalars and vectors, allowing representation of quantities in higher dimensions and more complex systems.

For instance, stress and strain in materials are described using tensor quantities.

Thus, tensors provide a general framework for representing physical quantities beyond simple scalars and vectors.

Explanation: This question examines the result of vector addition. When two vectors are added, the result follows the rules of vector algebra.

Vector addition considers both magnitude and direction. The resultant vector is obtained using methods like triangle law or parallelogram law.

Unlike scalar addition, vector addition produces a quantity that still retains direction. Thus, the result remains a vector.

For example, combining two displacements results in a NET displacement, which is also a vector.

Thus, the sum of two vectors always results in another vector.

Explanation: Multiplying a vector by a scalar changes its magnitude but does not affect its direction unless the scalar is negative.

The resulting quantity still has both magnitude and direction, so it remains a vector. The scalar acts as a scaling factor.

If the scalar is positive, the direction remains the same; if negative, the direction reverses.

For example, doubling a velocity vector increases its magnitude but keeps its direction unchanged.

Thus, the product of a scalar and a vector is always another vector.

Explanation: The cross product of two vectors produces a vector that is perpendicular to both original vectors. This is a fundamental operation in three-dimensional vector algebra.

The magnitude of the cross product depends on the sine of the angle between the vectors, while the direction is given by the right-hand rule.

This operation is widely used in Physics to calculate quantities like torque and angular momentum.

For example, rotating forces produce effects perpendicular to the plane of action.

Thus, the cross product always results in a vector quantity.

Explanation: Scalar quantities are defined as those physical quantities that have only magnitude and no associated direction. This question focuses on distinguishing such quantities from vectors.

Common scalar quantities include length, mass, time, temperature, and energy. These can be completely described using a numerical value and unit alone, without any reference to direction.

In contrast, quantities like velocity, force, and acceleration require direction for their complete description, making them vectors.

For example, saying “5 meters” fully describes a length, but “5 m/s” is incomplete without direction.

Thus, scalar quantities are those that are independent of direction and described entirely by magnitude.

Explanation: Vector quantities are physical quantities that require both magnitude and direction for complete description. This question aims to identify such a quantity.

Examples include force, velocity, acceleration, and Electric Field. These quantities change if their direction changes, even if their magnitude remains the same.

Scalar quantities like mass or distance do not depend on direction and therefore are not vectors.

For instance, force applied in different directions produces different effects, which highlights its vector nature.

Thus, a quantity that depends on both magnitude and direction is classified as a vector.

Explanation: Scalars are quantities that have only magnitude. When two scalars are multiplied, the result follows the rules of ordinary algebra.

Since neither quantity has direction, their product also does not involve any directional component. The result is simply another numerical value with appropriate units.

This is different from vector multiplication, where direction plays a role and may produce either scalar or vector results depending on the operation.

For example, multiplying two numbers like 5 and 3 gives 15, which is still a scalar.

Thus, the multiplication of two scalar quantities always results in another scalar.

Explanation: Work done by a force is calculated using the dot product between the force vector and the displacement vector. It measures how much of the force contributes in the direction of motion.

The dot product is found by multiplying corresponding components of the vectors and adding them together. This results in a scalar quantity.

If the force and displacement are in similar directions, work is positive; if perpendicular, work is zero; and if opposite, it becomes negative.

Imagine pushing an object—only the component of force in the direction of movement contributes to work.

Thus, by applying the dot product between force and displacement, the work done is determined.

Explanation: The equation of a plane can be determined using a normal vector and a point through which the plane passes. Here, the given vector acts as the normal to the plane.

The general form of the plane equation is obtained using the normal vector components and coordinates of the given point. This involves substituting these values into the standard plane equation.

The resulting equation represents all points that satisfy the condition of being perpendicular to the given vector.

For example, a flat surface can be defined if its orientation and a point on it are known.

Thus, using the normal vector and a point, the equation of the plane is formed.

Explanation: Work done in moving a particle under a constant force is calculated using the dot product of force and displacement. The displacement is obtained by subtracting initial position from final position.

First, the displacement vector is found as r₂ − r₁. Then, the dot product of this displacement with the force vector is calculated.

This gives a scalar value representing the work done. The direction of displacement relative to force determines whether the work is positive or negative.

For example, pushing an object in the direction of motion increases work done.

Thus, by computing displacement and applying the dot product with force, the work done is determined.

Explanation: Two vectors are perpendicular if their dot product is zero. This condition is used to find the unknown component in one of the vectors.

The dot product is calculated by multiplying corresponding components and adding them. Setting this result equal to zero gives an equation involving the unknown a.

Solving this equation yields the required value. Proper handling of signs and coefficients is essential during calculation.

For example, perpendicular directions like north and east have zero dot product.

Thus, by applying the perpendicular condition, the unknown value is determined.

Explanation: This problem involves finding magnitudes of two forces using different resultant conditions. When forces act in opposite directions, their resultant is the difference of their magnitudes.

When the same forces act at 90°, the resultant is found using the formula R² = A² + B².

Using both conditions, two equations are formed. Solving these simultaneously gives the magnitudes of the forces.

Think of forces combining differently depending on their angle—opposite directions reduce effect, while perpendicular directions combine geometrically.

Thus, by applying both conditions and solving equations, the original forces are determined.

Explanation: This question uses vector relationships involving magnitudes and differences. The magnitude of a vector difference depends on the angle between the vectors.

The formula |A − B|² = |A|² + |B|² − 2AB cosθ is used. Given that |A − B| equals |C|, this condition leads to a relationship involving cosθ.

By comparing both sides and simplifying, a trigonometric condition is obtained. Solving it gives the angle between the vectors.

For example, the relative orientation of two vectors determines how their difference behaves.

Thus, by using magnitude relations and trigonometry, the angle between vectors is determined.

Explanation: This problem relates magnitudes of vectors and their difference to the angle between them. The magnitude of A − B depends on both magnitudes and the angle between vectors.

Using the identity |A − B|² = |A|² + |B|² − 2AB cosθ, the given condition allows simplification of the equation.

By equating |A| and |A − B|, terms cancel out, leading to a relation involving |B| and cosθ. This helps determine the required condition.

For example, geometric relations between vectors can constrain their magnitudes and angles.

Thus, by applying vector identities and simplifying, the required relationship is obtained.

Explanation: The magnitude of the resultant of two forces depends on both their magnitudes and the angle between them. The general formula is R = √(A² + B² + 2AB cosθ).

The maximum resultant occurs when both forces act in the same direction (θ = 0°), giving R = A + B. The minimum occurs when they act in opposite directions (θ = 180°), giving R = |A − B|.

Thus, the resultant must always lie between these two extreme values. Any value outside this range is not possible.

For example, if two people pull an object, the combined effect cannot exceed the sum of their forces or be less than their difference.

Thus, by identifying the possible range of resultant values, we can determine which value is not feasible.

Explanation: This question involves comparing magnitudes of the sum and difference of two vectors. These magnitudes depend on the angle between the vectors.

Using identities, |A + B|² = |A|² + |B|² + 2AB cosθ and |A − B|² = |A|² + |B|² − 2AB cosθ.

Equating both expressions leads to cancellation of common terms, leaving a condition involving cosθ. This simplifies to cosθ = 0.

When cosθ = 0, the angle between the vectors is 90°, meaning they are perpendicular.

Thus, equality of magnitudes of sum and difference implies perpendicular vectors.

Explanation: This question involves calculating the cross product of two vectors in two dimensions. The result of a cross product is a vector perpendicular to the plane containing the original vectors.

In this case, both vectors lie in the XY-plane, so their cross product will be along the Z-axis. The calculation involves forming a determinant using unit vectors and the components of the given vectors.

The magnitude is obtained by multiplying components in a specific order and subtracting accordingly. The direction is determined using the right-hand rule.

For example, rotating one vector toward another determines the direction of the resulting vector along the perpendicular axis.

Thus, by applying the determinant method and right-hand rule, the cross product is obtained as a vector along the Z-axis.