Vector Algebra

01

PYQ 2023

medium

mathematics ID: ap-eapce

Let and be two vectors. If is a vector such that, then is:

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02

PYQ 2023

medium

mathematics ID: ap-eapce

If and are the position vectors of three points A, B, and C respectively, then is:

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is a right-angled triangle

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is an isosceles triangle

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are collinear points

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is a scalene triangle

03

PYQ 2023

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mathematics ID: ap-eapce

If and are two vectors such that is minimum, then

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04

PYQ 2023

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mathematics ID: ap-eapce

If,,, then

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05

PYQ 2023

medium

mathematics ID: ap-eapce

Planes and are defined by vectors. If and is parallel to their intersection, then

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PYQ 2023

medium

mathematics ID: ap-eapce

Let be three non-coplanar vectors. Then the point of intersection of the line joining the points, and the line joining the points, is

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07

PYQ 2023

medium

mathematics ID: ap-eapce

$If,, and are three vectors, then a vector which is perpendicular to and is$

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Official Solution

Correct Option: (4)

$We are given three vectors,, and . We are asked to find a vector that is perpendicular to both and . Step 1: Compute the cross product . The cross product is calculated as follows: Thus, . Step 2: Find a vector perpendicular to and . Since, we need a vector that is perpendicular to both and . The required vector is: First, calculate, which is the same as the previous result: Now calculate the cross product of and : Thus, the correct answer is .$

08

PYQ 2023

medium

mathematics ID: ap-eapce

$Let be two vectors, and be a vector along the vector such that . If the direction of is opposite to that of, then$

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Official Solution

Correct Option: (3)

$We are given: Compute: Now, since the direction of is **opposite**, we take: Given, we compute: Hence,$

09

PYQ 2023

medium

mathematics ID: ap-eapce

$If is the position vector of the centroid of triangle and are respectively the position vectors of its vertices and, then$

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Official Solution

Correct Option: (2)

$Let: Let centroid: We know: But using vectors: Then using identities or direct vector calculation (from solution steps), the result simplifies to:$

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PYQ 2023

medium

mathematics ID: ap-eapce

$If are vectors such that, then the sum of the values of$

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Official Solution

Correct Option: (3)

$Magnitude: Sum of values:$

11

PYQ 2023

medium

mathematics ID: ap-eapce

$If are two vectors, then the component vector of perpendicular to is$

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Official Solution

Correct Option: (3)

$Component of perpendicular to is given by:$

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PYQ 2023

medium

mathematics ID: ap-eapce

$Let be two vectors and be a vector in the plane of . If is orthogonal to and the magnitude of is, then is$

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Official Solution

Correct Option: (4)

$Since lies in the plane of and, write: Given: and Solving this system (not shown due to complexity) leads to:$

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PYQ 2023

medium

mathematics ID: ap-eapce

$Let be the line passing through the points and, and be the plane passing through the point and perpendicular to the vector . If is the angle between and, then :$

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Official Solution

Correct Option: (1)

$We are given that is the line passing through the points and, and is the plane passing through the point and perpendicular to the vector . The angle between the line and the plane is found using the formula involving the direction ratios of the line and the normal to the plane. We calculate using the dot product between the direction vector of the line and the normal vector of the plane, leading to the answer .$

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PYQ 2023

medium

mathematics ID: ap-eapce

$If,, and, then a unit vector perpendicular to the plane containing is:$

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Official Solution

Correct Option: (1)

$The normal vector to the plane is obtained by taking the cross product of the vectors and . After calculating the cross product, we normalize the resulting vector to obtain the unit vector perpendicular to the plane. This results in .$

15

PYQ 2023

medium

mathematics ID: ap-eapce

$Let,, and be the position vectors of three points A, B, C respectively. If G is the centroid of triangle ABC, then find:$

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Official Solution

Correct Option: (1)

$We are given the position vectors,, and of points A, B, and C. We are asked to compute, where G is the centroid of triangle ABC. Step 1: Calculate the centroid G of the triangle. The position vector of the centroid is the average of the position vectors of the points A, B, and C: Substituting the given vectors: Simplify the components: So, . Step 2: Calculate the squared distances,, and . The squared distance between two points is given by: For, we have: Similarly, for and : Step 3: Calculate . Step 4: Final Calculation. Now sum the values: Thus, the final answer is .$

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PYQ 2023

medium

mathematics ID: ap-eapce

$Let be the plane determined by the vectors and . Let be the plane determined by the vectors and . If is the angle between and, then$

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Official Solution

Correct Option: (2)

$We are given two planes and with the normal vectors and respectively. The angle between the planes is the same as the angle between their normal vectors. To find, we use the formula: Step 1: Compute the dot product : Step 2: Compute the magnitudes of the normal vectors: Step 3: Calculate : Thus, the angle between the planes is, which matches the correct answer.$

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PYQ 2023

medium

mathematics ID: ap-eapce

$Let be any three non-coplanar vectors. If are scalars such that and, then :$

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$0$

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Official Solution

Correct Option: (3)

$By using the given vector relations and applying vector operations, we can find that .$

18

PYQ 2023

medium

mathematics ID: ap-eapce

$Let be three vectors such that is perpendicular to and is perpendicular to . If and, then the angle between and is$

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Official Solution

Correct Option: (2)

$Step 1: Use perpendicularity to establish dot products. Step 2: Square the magnitude of the sum of vectors. Step 3: Expand the squared magnitude. Step 4: Substitute given values. Step 5: Solve for .} Step 6: Use the dot product formula to find the angle . Step 7: Determine the angle . Step 8: Conclusion. The angle between and is .$

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PYQ 2023

medium

mathematics ID: ap-eapce

$Let,, and be four vectors. If, then$

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$2$

Official Solution

Correct Option: (2)

$Step 1: Calculate the cross products. Step 2: Substitute into the equation for .} Step 3: Equate coefficients of .} Step 4: Solve for . Step 5: Conclusion. The value of is .$

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PYQ 2023

medium

mathematics ID: ap-eapce

$Let ABCD be a parallelogram and, and be the position vectors of the vertices A, B, D respectively. Then the position vector of one of the points of trisection of the diagonal AC is$

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Official Solution

Correct Option: (2)

$Step 1: Find the position vector of vertex C. Step 2: Find the position vectors of the trisection points of AC. For trisection point P dividing AC in ratio 1:2: For trisection point Q dividing AC in ratio 2:1: Step 3: Compare with the options. The position vector matches option (2). Step 4: Conclusion. The position vector of one of the points of trisection of the diagonal AC is .$

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PYQ 2024

medium

mathematics ID: ap-eapce

$Let be non-coplanar vectors. If,, then$

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Official Solution

Correct Option: (4)

$We are given the vector equations: Step 1: Express the sum of given vectors Adding both equations: Step 2: Finding the modulus Since the given vectors are non-coplanar, their sum must satisfy: Thus, the magnitude is:$

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PYQ 2024

medium

mathematics ID: ap-eapce

$Let be three unit vectors. Let, . If,,, and, then$

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Official Solution

Correct Option: (2)

$We are given the unit vectors and the equations: Step 1: Analyze the Given Conditions We know that, so: Expanding: Expanding using dot product properties: Since are unit vectors: Step 2: Solve for Using the equation, we equate the known dot product results: Substituting the given values: Solving for :$

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PYQ 2024

hard

mathematics ID: ap-eapce

$If are position vectors of 4 points such that, then the ratio in which the line joining divides the line segment joining and is:$

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Official Solution

Correct Option: (4)

$We are given the vector equation: Rearrange the equation to solve for : Thus, we can express as a linear combination of,, and, and the ratio in which the line joining divides the segment joining and can be found as: Thus, the correct answer is .$

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PYQ 2024

medium

mathematics ID: ap-eapce

$If are 3 vectors such that and, then the angle between the vectors and is:$

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Official Solution

Correct Option: (1)

$We are given the following conditions: Using the property, we can write . Now, to find the angle between and, we use the formula for the dot product: Substitute the known values and solve for, we find that: Thus, the correct answer is .$

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PYQ 2025

medium

mathematics ID: ap-eapce

$For, if the vectors, and are coplanar and, then the value of is:$

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Official Solution

Correct Option: (4)

$Step 1: Compute Step 2: Compute Step 3: Use given equation and solve for Now compute the RHS: Equating both: We can factor: Try polynomial division: So we can verify:$

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PYQ 2025

medium

mathematics ID: ap-eapce

$Points and are given by and . A line along the vector passes through point, and another line along the vector passes through point . If a line along the vector intersects both the lines along and at and respectively, then ?$

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Official Solution

Correct Option: (3)

$Let us define the parametric equations of the three lines: Line through and parallel to : Line through and parallel to : Line along : Let this line intersect the above two lines at points and, with parameter, so: Now, we equate this vector with each line to find the common point of intersection: Let From the equality:$

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PYQ 2025

medium

mathematics ID: ap-eapce

$is a point on the line parallel to the vector . is a point on the plane parallel to the vectors and . Then, the point of intersection of the line and the plane is:$

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Official Solution

Correct Option: (1)

$The line passes through the point and is parallel to the vector . So, the parametric form of the line is: The plane passes through point and is parallel to the vectors: So, any point on the plane can be written as: To find the intersection of the line and the plane, equate both expressions: Equating components: Substituting (i) and (ii) into (iii): Now substitute into the line equation: Multiply by -1:$

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PYQ 2025

medium

mathematics ID: ap-eapce

$If, and are three vectors such that, then$

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Official Solution

Correct Option: (1)

High-Yield Trend

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28 MCQs

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Vector Algebra

High-Yield Trend

Chapter Questions 28 MCQs

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Chapter Questions
28 MCQs