Vector Algebra — Class 12 Mathematics PYQs

If $\vec{a}=3\hat{i}+2\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+\hat{k}$, then which of the following is correct ?

(A) $\vec{a}\parallel \vec{b}$
(B) $\vec{a}\perp \vec{b}$
(C) $|\vec{b}|>|\vec{a}|$
(D) $|\vec{a}|=|\vec{b}|$

If $\vec{a}=\hat{i}-4\hat{j}+9\hat{k}$ and $\vec{b}=2\hat{i}-8\hat{j}+\lambda \hat{k}$ are two mutually parallel vectors, then $\lambda$ is equal to :

(A) $-18$
(B) $18$
(C) $-\frac{34}{9}$
(D) $\frac{34}{9}$

The unit vector $\perp$ to the vectors $\hat{i}-\hat{j}$ and $\hat{i}+\hat{j}$ is

(A) $\hat{k}$
(B) $-\hat{k}+\hat{j}$
(C) $\frac{\hat{i}-\hat{j}}{\sqrt{2}}$
(D) $\frac{\hat{i}+\hat{j}}{\sqrt{2}}$

If $\vec{a}+\vec{b}+\vec{c}=\vec{0}$, $|\vec{a}|=\sqrt{37}$, $|\vec{b}|=3$ and $|\vec{c}|=4$, then the angle between $\vec{b}$ and $\vec{c}$ is

(A) $\frac{\pi }{6}$
(B) $\frac{\pi }{4}$
(C) $\frac{\pi }{3}$
(D) $\frac{\pi }{2}$

The projection vector of vector $\vec{a}$ on vector $\vec{b}$ is

(A) $\frac{\vec{a}\cdot \vec{b}}{|\vec{b}{|}^2}\vec{b}$
(B) $\frac{\vec{a}\cdot \vec{b}}{|\vec{b}|}$
(C) $\frac{\vec{a}\cdot \vec{b}}{|\vec{a}|}$
(D) $\frac{\vec{a}\cdot \vec{b}}{|\vec{a}{|}^2}\vec{b}$

The values of $x$ for which the angle between the vectors $\vec{a}=2x^2\hat{i}+4x\hat{j}+\hat{k}$ and $\vec{b}=7\hat{i}-2\hat{j}+x\hat{k}$ is obtuse, is :

(A) $0$ or $\frac{1}{2}$
(B) $x>\frac{1}{2}$
(C) $\left(0,\frac{1}{2}\right)$
(D) $\left[0,\frac{1}{2}\right]$

If $\vec{p}$ and $\vec{q}$ are two unit vectors and $\alpha$ be the angle between them. Then $(\vec{p}+\vec{q})$ will be a unit vector for what value of $\alpha$ ?

(A) $\frac{\pi }{4}$
(B) $\frac{\pi }{3}$
(C) $\frac{\pi }{2}$
(D) $\frac{2\pi }{3}$

If projection of $\vec{a}=\alpha \hat{i}+\hat{j}+4\hat{k}$ on $\vec{b}=2\hat{i}+6\hat{j}+3\hat{k}$ is 4 units, then $\alpha$ is

(A) $-13$
(B) $-5$
(C) $13$
(D) $5$

If $|\vec{a}|=2$, $|\vec{b}|=3$ and $\vec{a}\cdot \vec{b}=4$, then evaluate $|\vec{a}+2\vec{b}|$.

The diagonals of a parallelogram are given by $\vec{a}=2\hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+3\hat{j}-\hat{k}$. Find the area of the parallelogram.