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Explore popular questions from Vectors and ThreeDimensional Geometry for JEE Advanced. This collection covers Vectors and ThreeDimensional Geometry previous year JEE Advanced questions hand picked by experienced teachers.

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Vectors and ThreeDimensional Geometry

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Q 1. If {tex} \vec { a } , \vec { b } , \vec { c } {/tex} are non coplanar unit vectors such that {tex} \vec { a } \times ( \vec { b } \times \vec { c } ) = \frac { ( \vec { b } + \vec { c } ) } { \sqrt { 2 } } , {/tex} then the angle between {tex} \vec { a } {/tex} and {tex} \vec { b } {/tex} is

{tex} \frac { 3 \pi } { 4 } {/tex}

B

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

C

{tex} \pi / 2 {/tex}

D

{tex} \pi {/tex}

Explanation

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Q 2. Let {tex} a = 2 i + j - 2 k {/tex} and {tex} b = i + j . {/tex} If {tex} c {/tex} is a vector such that {tex} a {/tex}. {tex} c = | \mathbf { c } | , | c - a | = 2 \sqrt { 2 } {/tex} and the angle between {tex} ( a \times b ) {/tex} and {tex} c {/tex} is {tex} 30 ^ { \circ } , {/tex} then {tex} | ( a \times b ) \times c | = {/tex}

A

{tex} 2 / 3 {/tex}

{tex} 3 / 2 {/tex}

C

{tex}2{/tex}

D

{tex}3{/tex}

Explanation

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Q 3. If {tex} \vec { a } = ( \hat { i } + \hat { j } + \hat { k } ) , \vec { a } \cdot \vec { b } = 1 {/tex} and {tex} \vec { a } \times \vec { b } = \hat { j } - \hat { k } , {/tex} then {tex} \vec { b } {/tex} is

A

{tex} \hat { i } - \hat { j } + \hat { k } {/tex}

B

{tex} 2 \hat { j } - \hat { k } {/tex}

{tex} \hat { i } {/tex}

D

{tex} 2 \hat { i } {/tex}

Explanation

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Q 4. If the lines
{tex} \frac { x - 1 } { 2 } = \frac { y + 1 } { 3 } = \frac { z - 1 } { 4 } {/tex} and {tex} \frac { x - 3 } { 1 } = \frac { y - k } { 2 } = \frac { z } { 1 } {/tex} intersect {tex} , {/tex} then
the value of {tex} k {/tex} is

A

{tex} 3 / 2 {/tex}

{tex} 9 / 2 {/tex}

C

{tex} - 2 / 9 {/tex}

D

{tex} - 3 / 2 {/tex}

Explanation


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Q 5. A plane {tex} 2 x + 3 y + 5 z = 1 {/tex} has a point {tex} P {/tex} which is at minimum distance from line joining {tex} A ( 1,0 , - 3 ) {/tex} and {tex} B ( 1 , - 5,7 ) {/tex} then distance {tex} A P {/tex} is equal to

A

{tex} 3 \sqrt { 5 } {/tex}

{tex} 2 \sqrt { 5 } {/tex}

C

{tex} 4 \sqrt { 5 } {/tex}

D

None of these

Explanation

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Q 6. Let a plane passes through origin and is parallel to line {tex} \frac { x - 1 } { 2 } = \frac { y + 3 } { - 1 } = \frac { z + 1 } { - 2 } {/tex} such that the distance between plane and line is {tex} 5 / 3 , {/tex} then the equation of plane is

{tex} 2 x + 2 y - z = 0 {/tex}

B

{tex} x - 2 y - 2 z = 0 {/tex}

C

{tex} x + 2 y + 2 z = 0 {/tex}

D

{tex} 2 x - 2 y + z = 0 {/tex}

Explanation


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Q 7. If the shortest distance between lines {tex} \vec { r } = \hat { i } + 2 \hat { j } + 3 \hat { k } + \lambda _ { 1 } {/tex}{tex} ( 2 \hat { i } + 3 \hat { j } + 4 \hat { k } ) {/tex} and {tex} \vec { r } = 2 \hat { i } + 4 \hat { j } + 5 \hat { k } + \lambda _ { 2 } ( 3 \hat { i } + 4 \hat { j } + 5 \hat { k } ) {/tex} is {tex} x {/tex}, then {tex} \cos ^ { - 1 } \cos \sqrt { 6 } x {/tex} is equal to

A

{tex} \frac { 1 } { 2 } {/tex}

B

{tex} 0{/tex}

{tex}1{/tex}

D

{tex} \pi {/tex}

Explanation

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Q 8. If {tex} \vec { r } = \lambda ( \vec { a } \times \vec { b } ) + \mu ( \vec { b } \times \vec { c } ) + \gamma ( \vec { c } \times \vec { a } ) {/tex} and {tex} [ \vec { a } \vec { b } \vec { c } ] = \frac { 1 } { 8 } , {/tex} then {tex} \lambda + \mu {/tex} {tex} + \gamma {/tex} is

A

{tex} 8 ( \vec { r } \cdot \vec { a } ) {/tex}

B

{tex} 8 ( \vec { r } \cdot \vec { b } ) {/tex}

C

{tex} 8 ( \vec { r } \cdot \vec { c } ) {/tex}

{tex} 8 \vec { r } \cdot ( \vec { a } + \vec { b } + \vec { c } ) {/tex}

Explanation

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Q 9. In a triangle {tex} O A B , E {/tex} is the mid-point of {tex} O B {/tex} and {tex} D {/tex} is a point on {tex} A B {/tex} such that {tex} A D: D B = 2: 1 . {/tex} If {tex} O D {/tex} and {tex} A E {/tex} intersect at {tex} P {/tex}, then ratio of {tex} \frac { O P } { P D } {/tex} is equal to

{tex}3:2{/tex}

B

{tex}2:3{/tex}

C

{tex}3:4{/tex}

D

{tex}4:3{/tex}

Explanation


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Q 10. If {tex} \vec { x } {/tex} and {tex} \vec { y } {/tex} be unit vectors and {tex} | \vec{z} | = \frac { 2 } { \sqrt { 7 } } {/tex} such that {tex}\vec z + \vec z \times \vec { x } = \vec { y } {/tex}, then the angle {tex} \theta {/tex} between {tex} \vec { x } {/tex} and {tex} \vec { z } {/tex} is

A

{tex} 30 ^ { \circ } {/tex}

{tex} 60 ^ { \circ } {/tex}

C

{tex} 90 ^ { \circ } {/tex}

D

None of these

Explanation

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Q 11. If a non-zero vector {tex} \vec { a } {/tex} is parallel to the line of intersection of the plane {tex} P _ { 1 } {/tex} determined by {tex} \hat { i } + \hat { j } {/tex} and {tex} \hat { i } - 2 \hat { j } {/tex} and plane {tex} P _ { 2 } {/tex} determined by vector {tex} 2 \hat { i } + \hat { j } {/tex} and {tex} 3 \hat { i } + 2 \hat { k } {/tex}, then angle between {tex} \vec { a } {/tex} and vector {tex} \hat { i } - 2 \hat { j } + 2 \hat { k } {/tex} is

A

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

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

C

{tex} \frac { \pi } { 3 } {/tex}

D

None of these

Explanation

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Q 12. {tex} \vec { a } {/tex} and {tex} \vec { b } {/tex} are non-zero, non-collinear vectors such that {tex} | \vec { a } | = {/tex} {tex} 2 ,\ \vec { a } \cdot \vec { b } = 1 {/tex} and angle between {tex} \vec { a } {/tex} and {tex} \vec { b } {/tex} is {tex} \frac { \pi } { 3 } {/tex}. If {tex} \vec { r } {/tex} is any vector satisfying {tex} \vec { r } \cdot \vec { a } = 2 ,\ \vec { r } \cdot \vec { b } = 8 ,\ ( \vec { r } + 2 \vec { a } - 10 \vec { b } ) ( \vec { a } \times \vec { b } ) = 4 \sqrt { 3 } {/tex} and is equal to {tex} \vec { r } + 2 \vec { a } - 10 \vec { b } = \lambda ( \vec { a } \times \vec { b } ) , {/tex} then {tex} \lambda = {/tex}

A

{tex} \frac { 1 } { 2 } {/tex}

2

C

{tex} \frac { 1 } { 4 } {/tex}

D

4

Explanation

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Q 13. Let {tex} \hat { a } , \hat b , \hat { c } {/tex} be unit vectors such that {tex} \hat { a } \times \hat { b } = \hat { c } {/tex} and {tex} \hat { a } \cdot \hat { b } = 0 {/tex}. Also, {tex} \vec { x } {/tex} is any vectors such that {tex} [\vec x\ \hat { b }\ \hat { c } ] = 3 , [\vec x\ \hat { c }\ \hat { a } ] = 4 {/tex} and {tex} [ \vec { x }\ \hat { a }\ \hat { b } ] = 2 {/tex}. Then {tex} \vec { x } {/tex} is equal to

A

{tex} 2 \hat { a } + 3 \hat { b } + \hat { c } {/tex}

{tex} 3 \hat { a } + 4 \hat { b } + 2 \hat { c } {/tex}

C

{tex} \hat { a } + 2 \hat { b } + 3 \hat { c } {/tex}

D

None of these

Explanation

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Q 14. If {tex} \vec { a } ,\ \vec { b } {/tex} and {tex} \vec { c } {/tex} are three non-coplanar unimodular vectors, each inclined with other at an angle {tex} 30 ^ { \circ } , {/tex} then volume of tetrahedron whose edges are {tex} \vec { a } ,\ \vec { b } {/tex} and {tex} \vec { c } {/tex} is

{tex} \frac { \sqrt { 3 \sqrt { 3 } - 5 } } { 12 } {/tex}

B

{tex} \frac { 3 \sqrt { 3 } + 5 } { 12 } {/tex}

C

{tex} \frac { 5 \sqrt { 2 } + 3 } { 12 } {/tex}

D

None of these

Explanation