# Question 1 Review Exercise 3

Solutions of Question 1 of Review Exercise 3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.

Chose the correct option.

i. The value of $\hat{i} \cdot(\hat{j} \times \hat{k})+\hat{j} \cdot(\hat{i} \times \hat{k})+\hat{k} \cdot(\hat{i} \times \hat{j})$

• (a) $0$
• (b) $1$
• (c) $1$
• (d) $3$
(a): $0$

ii. The vectors $3 \hat{i}+5 \hat{j}+2 \hat{k}$, $2 \hat{i}-3 \hat{j}-5 \hat{k}$ and $5 \hat{i}+2 \hat{j}-3 \hat{k}$ forms the sides of triangle which is:

• (a) Equilateral
• (b) Isosceles but not right angle
• (c) Right angled but not isosceles
• (d) right angled ind isosceles
(a): Equilateral

iii. Two vectors $\hat{i}-2 \hat{i}+\hat{j}+3 \hat{k}$, $\vec{b}+\hat{i} \cdot \lambda \hat{j} \cdot 6 \hat{k}$ are parallel it $\lambda=$

• (a) $2$
• (b) $-3$
• (c) $3$
• (d) $-2$
(d): $-2$

iv. If $\mid \vec{a}+\vec{b}=\vec{a}-\vec{b}$. then

• (a) $\vec{a} \| \vec{b}$
• (b) $\vec{a} \perp \vec{b}$
• (c) $|\vec{a}|=|\vec{b}|$
• (d) None of these
(b): $\vec{a} \perp \vec{b}$

v. The projection of the vector $2 \hat{i}+3 \hat{j}-2 \hat{k}$ on the vector $\hat{i}+2 \hat{j}+3 \hat{k}$ is:

• (a) $\dfrac{1}{\sqrt{14}}$
• (b) $\dfrac{2}{\sqrt{14}}$
• (c) $\dfrac{3}{14}$
• (d) None of these
(b): $\dfrac{2}{\sqrt{14}}$

vi. Find nun-zero scalar $\alpha . \beta$ for which $\alpha(\vec{a}+2 \vec{b})-\beta \vec{a}+(4 \vec{b}-\vec{a})=0$ for all vectors $\vec{a}$ and $\vec{b}$

• (a) $\alpha=-2, \beta=-3$
• (b) $\alpha=2 \cdot \beta=-3$
• (c) $\alpha=1 . \beta=-3$
• (d) $\alpha=-2, \beta=3$
(a): $\alpha=-2, \beta=-3$

vii. If $\vec{a} \cdot \vec{b} . \vec{c}$ are position vectors of the vertices of a $\angle A A B C$. then $\vec{B} \cdot B^2 \cdot(\vec{i}$

• (a) $0$
• (b) $2a$
• (c) $2c$
• (d) $3c$
(a): $0$

viii. If $\theta$ be the angel between any two vectors $\vec{a}$ and $\vec{b}$. then $\vec{a} \cdot \vec{b} \mid=$ $\vec{a} \times \vec{b}_i$, when $\theta$ is equal to

• (a) $0$
• (b) $\dfrac{\pi}{4}$
• (c) $\dfrac{\pi}{2}$
• (d) $\pi$
©: $\dfrac{\pi}{2}$