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JEE Main

Explore popular questions from Vectors for JEE Main. This collection covers Vectors previous year JEE Main questions hand picked by experienced teachers.

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Q 1. If {tex} \vec { a } , \vec { b } , \vec { c } {/tex} are vectors such that {tex} [ \vec { a } \quad \vec { b } \quad \vec { c } ] = 4 {/tex}
then {tex} \left[ \begin{array} { l l l } { \vec { a } \times \vec { b } } & { \vec { b } \times \vec { c } } & { \vec { c } \times \vec { a } } \end{array} \right] = {/tex}

16

B

64

C

4

D

8

Explanation


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Q 2. The vectors {tex} \overrightarrow { A B } = 3 \hat { i } + 4 \hat { k } {/tex} and {tex} \overrightarrow { A C } = 5 \hat { i } - 2 \hat { j } + 4 \hat { k } {/tex} are the sides of a triangle {tex} A B C {/tex}. The length of the median through {tex} A {/tex} is

A

{tex} \sqrt { 72 } {/tex}

{tex} \sqrt { 33 } {/tex}

C

{tex} \sqrt { 288 } {/tex}

D

{tex} \sqrt { 18 } {/tex}

Explanation


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Q 3. {tex} \vec { a } , \vec { b } , \vec { c } {/tex} are three vectors, such that {tex} \vec { a } + \vec { b } + \vec { c } = 0 , {/tex} where {tex} | \vec { a } | = 1 , \ | \vec { b } | = 2 , | \vec { c } | = 3 , \quad {/tex} then {tex} \quad \vec { a } \cdot \vec { b } + \vec { b } \cdot \vec { c } + \vec { c } \cdot \vec { a } \quad {/tex} is equal to

-7

B

7

C

1

D

0

Explanation

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Q 4. Let {tex} \vec { a } , \vec { b } {/tex} and {tex} \vec { c } {/tex} be three non-zero vectors such that no two of these are collinear. If the vector {tex} \vec { a } + 2 \vec { b } {/tex} is collinear with {tex} \vec { c } {/tex} and {tex} \vec { b } + 3 \vec { c } {/tex} is collinear with {tex} \vec { a } {/tex} ( {tex}\lambda {/tex} being some non-zero scalar) then {tex} \vec { a } + 2 \vec { b } + 6 \vec { c } {/tex} equals

A

{tex} \lambda \vec { c } {/tex}

B

{tex} \lambda \vec { b } {/tex}

C

{tex} \lambda \vec { a } {/tex}

0

Explanation


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Q 5. If {tex} \vec { a } , \vec { b } , \vec { c } {/tex} are non-coplanar vectors and {tex} \lambda {/tex} is a real number, then the vectors {tex} \vec { a } + 2 \vec { b } + 3 \vec { c } , \lambda \vec { b } + 4 \vec { c } {/tex} and {tex} ( 2 \lambda - 1 ) \vec { c } {/tex} are non-coplanar for

all except two values of {tex} \lambda {/tex}

B

all except one value of {tex} \lambda {/tex}

C

all values of {tex} \lambda {/tex}

D

no value of {tex} \lambda {/tex}

Explanation


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Q 6. Let {tex} \vec { a } , \vec { b } {/tex} and {tex} \vec { c } {/tex} be non-zero vectors such that {tex} ( \vec { a } \times \vec { b } ) \times \vec { c } = \frac { 1 } { 3 } | \vec { b } | | \vec { c } | \vec { a }{/tex} . If {tex} \theta {/tex} is the acute angle between the vectors {tex} b {/tex} and {tex} \vec { c } , {/tex} then {tex} \sin \theta {/tex} equals

A

{tex} \frac { 2 } { 3 } {/tex}

B

{tex} \frac { \sqrt { 2 } } { 3 } {/tex}

C

{tex} \frac { 1 } { 3 } {/tex}

{tex} \frac { 2 \sqrt { 2 } } { 3 } {/tex}

Explanation


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Q 7. Let {tex} \vec { a } = \hat { i } - \hat { k } , \vec { b } = x \hat { i } + \hat { j } + ( 1 - x ) \hat { k } {/tex} and {tex} \vec { c } = y \hat { i } + x \hat { j } + ( 1 + x - y ) \hat { k } {/tex}.
Then {tex} [ \vec { a } , \vec { b } , \vec { c } ] {/tex} depends on

A

only {tex} x {/tex}

B

only {tex} y {/tex}

neither {tex} x {/tex} nor {tex} y {/tex}

D

both {tex} x {/tex} and {tex} y {/tex}

Explanation



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Q 8. If {tex} ( \vec { a } \times \vec { b } ) \times \vec { c } = \vec { a } \times ( \vec { b } \times \vec { c } ) , {/tex} where {tex} \vec { a } , \vec { b } {/tex} and {tex} \vec { c } {/tex} are any three vectors such that {tex} \vec { a } \cdot \vec { b } \neq 0 , \vec { b } \cdot \vec { c } \neq 0 , {/tex} then {tex} \vec { a } {/tex} and {tex} \vec { c } {/tex} are

A

inclined at an angle of {tex} \pi / 3 {/tex} between them

B

inclined at an angle of {tex} \pi / 6 {/tex} between them

C

perpendicular

parallel

Explanation


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Q 9. The vector {tex} \vec { a } = \alpha \hat { i } + 2 \hat { j } + \beta \hat { k } {/tex} lies in the plane of the vectors {tex} \vec { b } = \hat { i } + \hat { j } {/tex} and {tex} \vec { c } = \hat { j } + \hat { k } {/tex} and bisects the angle between {tex} \vec { b } {/tex} and {tex} \vec { c } {/tex}. Then which one of the following gives possible values of {tex} \alpha {/tex} and {tex} \beta ? {/tex}

{tex} \alpha = 1 , \beta = 1 {/tex}

B

{tex} \alpha = 2 , \beta = 2 {/tex}

C

{tex} \alpha = 1 , \beta = 2 {/tex}

D

{tex} \alpha = 2 , \beta = 1 {/tex}

Explanation


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Q 10. Let {tex} \vec { u } = \hat { i } + \hat { j } , \vec { v } = \hat { i } - \hat { j } {/tex} and {tex} \vec { w } = \hat { i } + 2 \hat { j } + 3 \hat { k } . {/tex} If {tex} \hat { n } {/tex} is a unit vector such that {tex} \vec { u } \cdot \hat { n } = 0 {/tex} and {tex} \vec { v } \cdot \hat { n } = 0 {/tex}, then {tex} | \vec { w } \cdot \hat { n } | {/tex} is equal to

A

1

B

2

3

D

0

Explanation

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Q 11. The values of a, for which the points A, B, C with position vectors {tex} 2\vec{i} - \vec{j} + \vec{k}, \vec{i} - 3\vec{j}-5\vec{k} {/tex} and {tex} a \vec{i} - 3 \vec{j} + \vec{k} {/tex} respectively are the vertices of a right­-angled triangle at {tex}c{/tex} are

2 and 1

B

- 2 and - 1

C

- 2 and 1

D

2 and - 1

Explanation


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Q 12. If {tex} \hat { u } {/tex} and {tex} \hat { v } {/tex} are unit vectors and {tex} \theta {/tex} is the acute angle between them, then {tex} 2 \hat { u } \times 3 \hat { v } {/tex} is a unit vector for

A

no value of {tex} \theta {/tex}

exactly one value of {tex} \theta {/tex}

C

exactly two values of {tex} \theta {/tex}

D

more than two values of {tex} \theta {/tex}

Explanation


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Q 13. The non-zero vectors {tex} \vec { a } , \vec { b } {/tex} and {tex} \vec { c } {/tex} are related by {tex} \vec { a } = 8 \vec { b } {/tex} and {tex} \vec { c } = - 7 \vec { b } . {/tex} Then the angle between {tex} \vec { a } {/tex} and {tex} \vec { c } {/tex} is

{tex} \pi {/tex}

B

0

C

{tex} \frac { \pi } { 4 } {/tex}

D

{tex} \frac { \pi } { 2 } {/tex}

Explanation


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Q 14. If {tex} \vec { a } , \vec { b } , \vec { c } {/tex} are non-coplanar vector and {tex} \lambda {/tex} is a real number then {tex} \left[ \lambda ( \vec { a } + \vec { b } ) \lambda ^ { 2 } \vec { b } \lambda \vec { c } \right] = \left[ \begin{array} { c c c } { \vec { a } } & { \vec { b } + \vec { c } } & { \vec { b } } \end{array} \right] {/tex} for

no value of {tex} \lambda {/tex}

B

exactly one value of {tex} \lambda {/tex}

C

exactly two values of {tex} \lambda {/tex}

D

exactly three values of {tex} \lambda {/tex}

Explanation


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Q 15. Vector makes equal angles with x, y and z axis. Value of its components (in terms of magnitude of ) will be -

B

C

D

Explanation

Let the components of makes angles and with x, y and z axis respectively then

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Q 16. If a unit vector is represented by , then the value of 'c' is-

A

1

C

D

Explanation

Magnitude of unit vector = 1

By solving we get

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Q 17. The component of vector along the vector is-

B

C

D

5

Explanation

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Q 18. There are two force vectors, one of 5 N and other of 12 N at what angle the two vectors be added to get resultant vector of 17 N, 7 N and 13 N respectively-

and

B

and

C

and

D

, and

Explanation

For 17N both the vector should be parallel i.e. angle between them should be zero. For 7N both the vectors should be antiparallel i.e. angle between them should be 1800 For 13N both the vectors should be perpendicular to each other i.e. angle between them should be

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Q 19. If and then magnitude and direction of will be-

A

C

D

Explanation



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Q 20. Which pair of the following forces will never give resultant force of 2 N-

A

2 N and 2 N

B

1 N and 1 N

C

1 N and 3 N

1 N and 4 N

Explanation

If two vectors A and B are given then Range of their resultant can be written as . i.e. and
If B = 1 and A = 4 then their resultant will lies in between 3 N and 5 N. It can never be 2 N.

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Q 21. Two forces 3 N and 2 N are at an angle θ such that the resultant is R. The first force is now increased to 6 N and the resultant become 2R. The value of θ is-

A

B

C

Explanation

, then

...(i)
Now , then
...(ii)
from (i) and (ii) we get

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Q 22. Three concurrent forces of the same magnitude are in equilibrium. What is the angle between the forces? Also, name the triangle formed by the forces as sides-

equilateral triangle

B

equilateral triangle

C

an isosceles triangle

D

an obtuse angled triangle

Explanation

In N forces of equal magnitude works on a single point and their resultant is zero then angle between any two forces is given


If these three vectors are represented by three sides of triangle then they form equilateral triangle

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Q 23. If , then angle between and will be-

A

B

D

Explanation

Resultant of two vectors and can be given by

If then

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Q 24. If then the angle between and is-

A

π / 2

B

π / 3

π

D

π / 4

Explanation

If the angle between and is 0 or π Then .

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Q 25. If then which of the following statements is wrong-

A

B

C

Explanation

From the property of vector product, is perpendicular to both and and vector also, must lie in the plane formed by vector and . Thusmust be perpendicular to also but the cross product gives a vector which can not be perpendicular to itself. Thus the last statement is wrong.