EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 18 - Vector Algebra>JEE Advanced Paper 2 - 2016>Q 6

HARD

JEE Advanced

IMPORTANT

Earn 100

Let $\hat{u} = u_{1} \hat{i} + u_{2} \hat{j} + u_{3} \hat{k}$ be a unit vector in $R^{3}$ and $\hat{w} = \frac{1}{\sqrt{6}} (\hat{i} + \hat{j} + 2 \hat{k}) .$ Given that there exists a vector $\vec{v}$ in $R^{3}$ such that $|\hat{u} \times \vec{v}| = 1$ and $\hat{w} . (\hat{u} \times \vec{v}) = 1 .$ Which of the following statement(s) is (are) correct ?

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Important Questions on Vector Algebra

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 18 - Vector Algebra>JEE Advanced Paper 1 - 2021>Q 10

Let

\vec{u}, \vec{v}

and

\vec{w}

be vectors in three-dimensional space, where

\vec{u}

and

\vec{v}

are unit vectors which are not perpendicular to each other and

\vec{u} \cdot \vec{w} = 1, \vec{v} \cdot \vec{w} = 1, \vec{w} \cdot \vec{w} = 4

.
If the volume of the parallelopiped, whose adjacent sides are represented by the vectors

\vec{u}, \vec{v}

and

\vec{w}

, is

\sqrt{2}

, then the value of

| 3 \vec{u} + 5 \vec{v} |

is

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 18 - Vector Algebra>JEE Advanced Paper 1 - 2020>Q 11

In a triangle $P Q R$ , let $\vec{a} = \vec{Q R}, \vec{b} = \vec{R P}$ and $\vec{c} = \vec{P Q}$ .

If $|\vec{a}| = 3, |\vec{b}| = 4$ and $\frac{\vec{a} . (\vec{c} - \vec{b})}{\vec{c} . (\vec{a} - \vec{b})} = \frac{|\vec{a}|}{|\vec{a}| + |\vec{b}|}$ , then the value of ${|\vec{a} \times \vec{b}|}^{2}$ is _______

MEDIUM

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 18 - Vector Algebra>JEE Advanced Paper 1 - 2018>Q 12

Let

\vec{a}

and

\vec{b}

be two unit vectors such that

\vec{a} . \vec{b} = 0 .

For some

x, y \in R,

let

\vec{c} = x \vec{a} + y \vec{b} + (\vec{a} \times \vec{b}) .

If

|\vec{c}| = 2

and the vector

\vec{c}

is inclined at the same angle

α

to both

\vec{a}

and

\vec{b},

then the value of

8 \cos^{2} α

is

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 18 - Vector Algebra>JEE Advanced Paper 1 - 2015>Q 13

Let

Δ P Q R

be a triangle. Let

\vec{a} = \vec{Q R}, \vec{b} = \vec{R P} & \vec{c} = \vec{P Q}

. If

|\vec{a}| = 12, |\vec{b}| = 4 \sqrt{3}

and

\vec{b} \cdot \vec{c} = 24

, then which of the following is (are) true ?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 18 - Vector Algebra>JEE Advanced Paper 1 - 2014>Q 14

Let

\vec{x}, \vec{y}

and

\vec{z}

be three vectors each of magnitude

\sqrt{2}

and the angle between each pair of them is

\frac{π}{3} .

If

\vec{a}

is a non - zero vector perpendicular to

\vec{x}

and

\vec{y} \times \vec{z}

and

\vec{b}

is a non - zero vector perpendicular to

\vec{y}

and

\vec{z} \times \vec{x}

, then

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 18 - Vector Algebra>JEE Advanced Paper 1 - 2013>Q 16

Let

\vec{P R} = 3 \hat{i} + \hat{j} - 2 \hat{k} & \vec{S Q} = \hat{i} - 3 \hat{j} - 4 \hat{k}

determine diagonals of a parallelogram

P Q R S

and

\vec{P T} = \hat{i} + 2 \hat{j} + 3 \hat{k}

be another vector. Then the volume (in cubic unit) of the parallelepiped determined by the vectors

\vec{P T}, \vec{P Q} & \vec{P S}

is