Vector Algebra: CBSE Class 12 Previous Year Questions

The value of p for which vectors $\hat{i}+2\hat{j}+3\hat{k}$ and $2\hat{i}-p\hat{j}+\hat{k}$ are perpendicular to each other is

1. (A)0
2. (B)1
3. (C)$\frac{5}{2}$
4. (D)$-\frac{5}{2}$

The value of m for which the points with position vectors $-\hat{i}-\hat{j}+2\hat{k}$, $2\hat{i}+m\hat{j}+5\hat{k}$ and $3\hat{i}+11\hat{j}+6\hat{k}$ are collinear, is

1. (A)8
2. (B)$-8$
3. (C)2
4. (D)$\frac{5}{2}$

If $|\vec{a}|=8,|\vec{b}|=3$ and $|\vec{a}\times \vec{b}|=12$, then the value of $|\vec{a}\cdot \vec{b}|$

1. (A)$6\sqrt{3}$
2. (B)$8\sqrt{3}$
3. (C)$12\sqrt{3}$
4. (D)$3\sqrt{12}$

If $|\vec{a}|=5$ and $-2\le \lambda \le 1$, then the sum of greatest and the smallest value of $|\lambda \vec{a}|$ is

1. (A)$-5$
2. (B)5
3. (C)10
4. (D)15

Vector of magnitude 3 making equal angles with $x$ and y axes and perpendicular to z axis is

1. (A)$\hat{i}+2\sqrt{2}\hat{j}$
2. (B)$3\hat{k}$
3. (C)$\frac{3\sqrt{2}}{2}\hat{i}+\frac{3\sqrt{2}}{2}\hat{j}$
4. (D)$\sqrt{3}\hat{i}+\sqrt{3}\hat{j}+\sqrt{3}\hat{k}$

For two vectors $\vec{a}$ and $\vec{b}$
Assertion (A) : $|\vec{a}\times \vec{b}{|}^2+(\vec{a}\cdot \vec{b}{)}^2=|\vec{a}{|}^2|\vec{b}{|}^2$
Reason (R) : $|\vec{a}\times \vec{b}|=(\vec{a}\cdot \vec{b})\tan \theta$, $\left(\theta \ne \frac{\pi }{2}\right)$

1. (A)Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)Assertion (A) is false, but Reason (R) is true.

For any two vectors $\vec{a}$ and $\vec{b}$, which of the following statements is always true ?

1. (A)$\vec{a}\cdot \vec{b}\le |\vec{a}||\vec{b}|$
2. (B)$|\vec{a}+\vec{b}|\ge |\vec{a}|+|\vec{b}|$
3. (C)$|\vec{a}-\vec{b}|=|\vec{a}|-|\vec{b}|$
4. (D)$|\vec{a}\times \vec{b}|\ge |\vec{a}||\vec{b}|$

If $(\vec{a}+\vec{b})\cdot (\vec{a}-\vec{b})=198$ and $|\vec{a}|=10|\vec{b}|$, then :

1. (A)$|\vec{a}|=\sqrt{2}$
2. (B)$|\vec{b}|=\sqrt{2}$
3. (C)$|\vec{b}|=10\sqrt{2}$
4. (D)$|\vec{a}|=\frac{10}{\sqrt{2}}$

Assertion (A) : The vectors $\vec{a}$ and $(-2\vec{a})$, where $\vec{a}\ne \vec{0}$ are collinear vectors.
Reason (R) : $\vec{a}\cdot (-2\vec{a})=0$.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)Assertion (A) is false, but Reason (R) is true.

If $(3\hat{i}-2\hat{j}+5\hat{k})\times (4\hat{i}+p\hat{j}+q\hat{k})=\vec{0}$, then the values of p and q are :

1. (A)$p=-\frac{2}{3},q=\frac{5}{3}$
2. (B)$p=-\frac{8}{3},q=\frac{20}{3}$
3. (C)$p=\frac{20}{3},q=-\frac{8}{3}$
4. (D)$p=0,q=0$