Let $a, b$ be non-zero vectors. ``$a \cdot b = | a | | b |$'' is ``$a \parallel b$'' a

Let vectors $\mathrm { a }$ and $\mathrm { b }$ be non-parallel. If vector $\lambda a + b$ is parallel to $\boldsymbol { a } + 2 b$, then the real number $\lambda = $ $\_\_\_\_$.

2. Vector ${ } ^ { \mathbf { 1 } } = ( 2,4 )$ is collinear with vector ${ } ^ { \mathbf { 1 } } = ( x , 6 )$. Then the real number $x =$
(A) 2
(B) 3
(C) 4
(D) 6

Given vectors $a = ( 1,2 ) , b = ( 2 , - 2 ) , c = ( 1 , \lambda )$. If $c \parallel ( 2a + b )$, then $\lambda = $ $\_\_\_\_$.

Let vectors $\boldsymbol { a } = ( 1 , - 1 ) , \boldsymbol { b } = ( m + 1,2 m - 4 )$ . If $\boldsymbol { a } \perp \boldsymbol { b }$ , then $m =$ $\_\_\_\_$.

Given unit vectors $\boldsymbol { a } , \boldsymbol { b }$ with an angle of $45 ^ { \circ }$ between them, and $k \boldsymbol { a } - \boldsymbol { b}$ is perpendicular to $\boldsymbol { a }$ , then $k = $ $\_\_\_\_$.

14. Given vectors $\vec{a} = (3,1)$, $\vec{b} = (1,0)$, $\vec{c} = \vec{a} + k\vec{b}$. If $\vec{a} \perp \vec{c}$, then $k = \_\_\_\_$.

Given vectors $\boldsymbol { a } = ( m , 3 ) , \boldsymbol { b } = ( 1 , m + 1 )$ . If $\boldsymbol { a } \perp \boldsymbol { b }$ , then $m =$ $\_\_\_\_$ .

Given vectors $\boldsymbol { a } = ( 0,1 ) , \boldsymbol { b } = ( 2 , x )$ , if $\boldsymbol { b } \perp ( \boldsymbol { b } - 4 \boldsymbol { a } )$ , then $x =$
A. $- 2$
B. $- 1$
C. $1$
D. $2$

Given plane vectors $\vec{a} = (x, 1)$, $\vec{b} = (x-1, 2x)$. If $\vec{a} \perp (\vec{a} - \vec{b})$, then $|\vec{a}| = $ \_\_\_\_

Let $\hat{a}$ and $\hat{b}$ be two unit vectors. If the vectors $\vec{c} = \hat{a} + 2\hat{b}$ and $\vec{d} = 5\hat{a} - 4\hat{b}$ are perpendicular to each other, then the angle between $\hat{a}$ and $\hat{b}$ is
(1) $\frac{\pi}{6}$
(2) $\frac{\pi}{2}$
(3) $\frac{\pi}{3}$
(4) $\frac{\pi}{4}$

Let $\vec { u }$ be a vector coplanar with the vectors $\vec { a } = 2 \hat { i } + 3 \hat { j } - \widehat { k }$ and $\vec { b } = \hat { j } + \widehat { k }$. If $\vec { u }$ is perpendicular to $\vec { a }$ and $\vec { u } \cdot \vec { b } = 24$, then $| \vec { u } | ^ { 2 }$ is equal to:
(1) 84
(2) 336
(3) 315
(4) 256

Let $\vec { a } = 2 \hat { i } + \lambda _ { 1 } \hat { j } + 3 \hat { k } , \vec { b } = 4 \hat { i } + \left( 3 - \lambda _ { 2 } \right) \hat { j } + 6 \hat { k }$ and $\vec { c } = 3 \hat { i } + 6 \hat { j } + \left( \lambda _ { 3 } - 1 \right) \hat { k }$ be three vectors such that $\vec { b } = 2 \vec { a }$ and $\vec { a }$ is perpendicular to $\vec { c }$. Then a possible value of $\left( \lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } \right)$ is
(1) $\left( - \frac { 1 } { 2 } , 4,0 \right)$
(2) $( 1,5,1 )$
(3) $\left( \frac { 1 } { 2 } , 4 , - 2 \right)$
(4) $( 1,3,1 )$

Let $O$ be the origin. Let $\overrightarrow { O P } = x \hat { i } + y \hat { j } - \widehat { k }$ and $\overrightarrow { O Q } = - \hat { i } + 2 \hat { j } + 3 x \hat { k } , x , y \in R , x > 0$, be such that $| \overrightarrow { P Q } | = \sqrt { 20 }$ and the vector $\overrightarrow { O P }$ is perpendicular to $\overrightarrow { O Q }$. If $\overrightarrow { O R } = 3 \hat { i } + \mathrm { z } \hat { j } - 7 \hat { k } , z \in R$, is coplanar with $\overrightarrow { O P }$ and $\overrightarrow { O Q }$, then the value of $x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ is equal to
(1) 7
(2) 9
(3) 2
(4) 1

If the projection of $2\hat{i} + 4\hat{j} - 2\hat{k}$ on $\hat{i} + 2\hat{j} + \alpha\hat{k}$ is zero. Then, the value of $\alpha$ will be

Vectors $a \hat { i } + b \hat { j } + \hat { k }$ and $2 \hat { i } - 3 \hat { j } + 4 \hat { k }$ are perpendicular to each other when $3 a + 2 b = 7$, the ratio of $a$ to $b$ is $\frac { x } { 2 }$. The value of $x$ is $\_\_\_\_$ .

If $\vec { a }$ and $\vec { b }$ makes an angle $\cos ^ { - 1 } \left( \frac { 5 } { 9 } \right)$ with each other, then $| \vec { a } + \vec { b } | = \sqrt { 2 } | \vec { a } - \vec { b } |$ for $| \vec { a } | = n | \vec { b } |$ The integer value of n is $\_\_\_\_$

For $\lambda > 0$, let $\theta$ be the angle between the vectors $\vec { a } = \hat { i } + \lambda \hat { j } - 3 \hat { k }$ and $\vec { b } = 3 \hat { i } - \hat { j } + 2 \hat { k }$. If the vectors $\vec { a } + \vec { b }$ and $\vec { a } - \vec { b }$ are mutually perpendicular, then the value of $( 14 \cos \theta ) ^ { 2 }$ is equal to
(1) 50
(2) 40
(3) 25
(4) 20

Let $\vec { a }$ and $\vec { b }$ be two unit vectors such that the angle between them is $\frac { \pi } { 3 }$. If $\lambda \vec { a } + 2 \vec { b }$ and $3 \vec { a } - \lambda \vec { b }$ are perpendicular to each other, then the number of values of $\lambda$ in $[ - 1,3 ]$ is :
(1) 2
(2) 1
(3) 0
(4) 3

Let $O$, $A$, $B$ be three non-collinear points on the coordinate plane, where the vector $\overrightarrow{OA}$ is perpendicular to $\overrightarrow{OB}$. If points $C$ and $D$ are on the line $AB$ satisfying $\overrightarrow{OC} = \frac{3}{5}\overrightarrow{OA} + \frac{2}{5}\overrightarrow{OB}$, $3\overline{AD} = 8\overline{BD}$, and $\overrightarrow{OC}$ is perpendicular to $\overrightarrow{OD}$, then $\frac{\overline{OB}}{\overline{OA}} = $ (Express as a fraction in lowest terms)