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Vector Algebra Set-3 MCQ Online Practice Test | Class 12 | Quiz Tweek

Vector Algebra Set-3

Practice Important MCQs for Vector Algebra — Mathematics, Class 12.

Vector Algebra is central to many board and competitive problems because it converts geometric ideas (perpendicularity, coplanarity, distances, angles, areas) into algebraic conditions using dot products, cross products, and vector identities. Mastering these techniques lets you solve higher-level questions quickly and accurately with minimal computation.

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Q1. Let $\vec a=3\mathbf i-\mathbf j+2\mathbf k$ and $\vec b=\mathbf i+2\mathbf j-\mathbf k$ . Find all real $\lambda$ such that $\vec a+\lambda\vec b$ is perpendicular to $\vec a-\lambda\vec b$ .

(A)

$\displaystyle \lambda=\pm\sqrt{\frac{3}{7}}$

(B)

$\displaystyle \lambda=\pm\sqrt{\frac{7}{3}}$

(C)

$\displaystyle \lambda=\pm\sqrt{\frac{7}{12}}$

(D)

$\displaystyle \lambda=0$

Q2. The position vectors of the vertices $A,B,C$ of a triangle are $\vec a=(1,2,3)$ , $\vec b=(4,0,1)$ and $\vec c=(2,5,-1)$ . The foot $P$ of the perpendicular from $A$ to the line $BC$ is given by which point?

(A)

$\displaystyle \left(\frac{28}{11},\frac{20}{11},\frac{3}{11}\right)$

(B)

$\displaystyle \left(\frac{36}{11},\frac{10}{11},\frac{3}{11}\right)$

(C)

$\displaystyle \left(\frac{36}{11},\frac{20}{11},\frac{8}{11}\right)$

(D)

$\displaystyle \left(\frac{36}{11},\frac{20}{11},\frac{3}{11}\right)$

Q3. For which value of $k$ are the vectors $\vec u=(1,2,3)$ , $\vec v=(2,-1,4)$ and $\vec w=(k,1,-2)$ coplanar?

(A)

$\displaystyle -\frac{12}{11}$

(B)

$\displaystyle -\frac{11}{12}$

(C)

$\displaystyle -\frac{24}{11}$

(D)

$\displaystyle \frac{12}{11}$

Q4. Let $L_1$ be the line through $A(1,2,-1)$ and $B(2,0,3)$ , and $L_2$ the line through $C(0,1,2)$ and $D(3,-1,0)$ . The shortest distance between the skew lines $L_1$ and $L_2$ equals

(A)

$\displaystyle \frac{7}{2\sqrt{89}}$

(B)

$\displaystyle \frac{14}{\sqrt{89}}$

(C)

$\displaystyle \frac{7}{\sqrt{89}}$

(D)

$\displaystyle \frac{1}{\sqrt{89}}$

Q5. Let $\mathbf u,\mathbf v,\mathbf w$ be unit vectors with $\mathbf u\cdot\mathbf v=\tfrac12$ , $\mathbf v\cdot\mathbf w=\tfrac12$ and $\mathbf u\cdot\mathbf w=0$ . The angle $\theta$ between $\mathbf u\times\mathbf v$ and $\mathbf v\times\mathbf w$ is

(A)

$\displaystyle \cos^{-1}\!\left(\tfrac{1}{2}\right)$

(B)

$\displaystyle \cos^{-1}\!\left(\tfrac{1}{3}\right)$

(C)

$\displaystyle \cos^{-1}\!\left(\tfrac{2}{3}\right)$

(D)

$\displaystyle \frac{\pi}{2}$

Q6. If vectors $\mathbf{a}$ and $\mathbf{b}$ satisfy $|\mathbf{a}|=3$ , $|\mathbf{b}|=4$ and the angle between them is $60^\circ$ , then $|\mathbf{a}+\mathbf{b}|$ equals:

(A)

$5$

(B)

$7$

(C)

$\sqrt{37}$

(D)

$\sqrt{13}$

Q7. Let non-zero vectors $\mathbf{a}$ and $\mathbf{b}$ satisfy $|\mathbf{a}|=3$ , $|\mathbf{b}|=5$ . Find all real $\lambda$ such that $(\mathbf{a}+\lambda\mathbf{b})\perp(\mathbf{a}-\lambda\mathbf{b})$ .

(A)

$\dfrac{3}{5}$

(B)

$\pm\dfrac{3}{5}$

(C)

$\pm\dfrac{4}{5}$

(D)

$\pm\dfrac{5}{3}$

Q8. Let $\mathbf{a},\mathbf{b}$ be non-zero vectors with $|\mathbf{a}|=2$ and $|\mathbf{b}|=3$ . If $(\mathbf{a}-2\mathbf{b})\cdot(2\mathbf{a}+\mathbf{b})=0$ , then the cosine of the angle $\theta$ between $\mathbf{a}$ and $\mathbf{b}$ equals:

(A)

$-\dfrac{5}{9}$

(B)

$\dfrac{5}{9}$

(C)

$-\dfrac{10}{9}$

(D)

$\dfrac{10}{9}$

Q9. The shortest distance between the skew lines $L_1: \mathbf{r}=(1,2,3)+s(1,-1,2)$ and $L_2: \mathbf{r}=(2,0,1)+t(2,1,-1)$ is:

(A)

$\dfrac{7}{\sqrt{35}}$

(B)

$\dfrac{\sqrt{35}}{17}$

(C)

$\dfrac{5}{\sqrt{35}}$

(D)

$\dfrac{17}{\sqrt{35}}$

Q10. Let $\mathbf{a}$ and $\mathbf{b}$ be non-zero, non-parallel vectors in $\mathbb{R}^3$ . The set of all vectors $\mathbf{x}$ satisfying $\mathbf{x}\times(\mathbf{a}\times\mathbf{b})=\mathbf{a}$ is:

(A)

A unique vector $\mathbf{x}$ exists

(B)

All $\mathbf{x}$ such that $\mathbf{x}\cdot\mathbf{a}=1$ and $\mathbf{x}\cdot\mathbf{b}=0$

(C)

All $\mathbf{x}$ such that $\mathbf{x}\cdot\mathbf{a}=0$ and $\mathbf{x}\cdot\mathbf{b}=1$

(D)

No such vector exists

(A)

$\sqrt{14}$

(B)

$2\sqrt{7}$

(C)

$2\sqrt{5}$

(D)

$2\sqrt{3}$

Q12. If $\vec a=(1,2,3)$ , $\vec b=(4,\lambda,6)$ and $\vec c=(7,8,9)$ are coplanar, then the value of $\lambda$ is

(A)

$-5$

(B)

$10$

(C)

$0$

(D)

$5$

Q13. Let non-zero vectors $\vec a,\vec b$ satisfy $\vec a\cdot\vec b=0$ . The general solution of $\vec x\times\vec a=\vec b$ is

(A)

$\displaystyle \frac{\vec a\times\vec b}{|\vec a|^2}+\lambda\vec a,\ \lambda\in\mathbb{R}$

(B)

$\displaystyle \frac{\vec b\times\vec a}{|\vec a|^2}+\lambda\vec a,\ \lambda\in\mathbb{R}$

(C)

$\displaystyle \frac{\vec a\times\vec b}{2|\vec a|^2}+\lambda\vec a,\ \lambda\in\mathbb{R}$

(D)

$\displaystyle \frac{\vec a\times\vec b}{|\vec b|^2}+\lambda\vec a,\ \lambda\in\mathbb{R}$

Q14. Let $\vec a,\vec b,\vec c$ be three non-coplanar vectors in $\mathbb{R}^3$ . If real numbers $x,y,z$ satisfy $x(\vec a\times\vec b)+y(\vec b\times\vec c)+z(\vec c\times\vec a)=\vec 0$ , then which of the following is necessarily true?

(A)

$x+y+z=0$

(B)

$x=y=z$

(C)

$x=y=z=0$

(D)

At least one of $x,y,z$ is zero

Q15. The vertices of triangle $ABC$ are at $\vec A=(1,2,3)$ , $\vec B=(4,0,1)$ and $\vec C=(2,-1,2)$ . The area of $\triangle ABC$ is

(A)

$\sqrt{33}$

(B)

$\dfrac{\sqrt{66}}{2}$

(C)

$\sqrt{66}$

(D)

$\dfrac{\sqrt{11}}{2}$

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