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Vector Algebra Set-4 MCQ Online Practice Test | Class 12 | Quiz Tweek

Vector Algebra Set-4

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

Vector Algebra is a cornerstone chapter for Class 12 and competitive exams because it connects geometry with algebra using dot product, cross product, planes/lines in 3D, and key ideas like perpendicularity, coplanarity, skew lines distance, and minimum-norm vectors. Mastery of these results directly improves accuracy in both board examinations and entrance-level problem solving.

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Q1. Let $\mathbf{a}$ and $\mathbf{b}$ be vectors with $|\mathbf{a}|=3,\;|\mathbf{b}|=4$ and $|\mathbf{a}+\mathbf{b}|=5$ . Then $|\mathbf{a}-\mathbf{b}|=$

(A)

$1$

(B)

$\sqrt{41}$

(C)

$5$

(D)

$\sqrt{13}$

Q2. The line $L_1$ passes through the points $(1,0,2)$ and $(2,1,3)$ and the line $L_2$ passes through the points $(0,2,0)$ and $(1,-1,0)$ . The shortest distance between $L_1$ and $L_2$ is

(A)

$\dfrac{5}{\sqrt{6}}$

(B)

$\dfrac{5}{\sqrt{3}}$

(C)

$\dfrac{\sqrt{6}}{5}$

(D)

$\dfrac{7}{\sqrt{26}}$

Q3. Let $A(1,0,1),\;B(0,1,2),\;C(2,1,0)$ determine a plane $\pi$ and let $D(3,2,3)$ . The acute angle between the line $AD$ and the plane $\pi$ equals

(A)

$\displaystyle \arctan\!\left(\frac{1}{\sqrt{2}}\right)$

(B)

$\displaystyle \arctan(\sqrt{3})$

(C)

$\displaystyle \arccos\!\left(\frac{\sqrt{2}}{3}\right)$

(D)

$\displaystyle \arctan(\sqrt{2})$

Q4. Let $\mathbf{a}=(1,2,3)$ and $\mathbf{b}=(2,-1,0)$ . Find vector $\mathbf{x}$ such that $\mathbf{x}\times\mathbf{a}=\mathbf{b}$ and $\mathbf{x}\cdot\mathbf{a}=11$ .

(A)

$(0,1,3)$

(B)

$(1,2,2)$

(C)

$(2,1,2)$

(D)

$(1,2,-2)$

Q5. Assertion (A): If non-zero vectors $\mathbf{u},\mathbf{v},\mathbf{w}$ satisfy $\mathbf{u}\times\mathbf{v}=\mathbf{v}\times\mathbf{w}=\mathbf{w}\times\mathbf{u}$ , then $\mathbf{u},\mathbf{v},\mathbf{w}$ are coplanar.
Reason (R): From $\mathbf{u}\times\mathbf{v}=\mathbf{v}\times\mathbf{w}$ we get $(\mathbf{u}-\mathbf{w})\times\mathbf{v}=\mathbf{0}$ , so $\mathbf{u}-\mathbf{w}$ is parallel to $\mathbf{v}$ .

(A)

Both A and R are true and R is a correct explanation of A.

(B)

Both A and R are true but R is NOT a correct explanation of A.

(C)

A is true but R is false.

(D)

A is false but R is true.

Q6. If vectors $\vec{a}=(2,-1,3)$ and $\vec{b}=(1,4,-2)$ represent adjacent sides of a parallelogram, then the area of the parallelogram is

(A)

( $\sqrt{115}$ )

(B)

( $15$ )

(C)

( $\sqrt{230}$ )

(D)

( $\dfrac{\sqrt{230}}{2}$ )

Q7. Let $\vec{a}=(1,2,3)$ , $\vec{b}=(4,0,-1)$ and $\vec{c}=(k,1,2)$ . If the three vectors are coplanar, the value of $k$ is

(A)

( $-\dfrac{3}{2}$ )

(B)

( $-3$ )

(C)

( $\dfrac{3}{2}$ )

(D)

( $-\dfrac{1}{2}$ )

Q8. If $|\vec{a}|=3$ , $|\vec{b}|=4$ and the angle between $\vec{a}$ and $\vec{b}$ is $60^\circ$ , then $|2\vec{a}-\vec{b}|$ equals

(A)

( $\sqrt{14}$ )

(B)

( $6$ )

(C)

( $\sqrt{13}$ )

(D)

( $2\sqrt{7}$ )

Q9. Let the lines be given by

L_1:\ \vec{r}=(1,2,3)+s(2,-1,1),\quad L_2:\ \vec{r}=(3,0,1)+t(1,1,2).

The shortest distance between $L_1$ and $L_2$ is

(A)

( $\dfrac{2}{3}$ )

(B)

( $\dfrac{2\sqrt{3}}{3}$ )

(C)

( $\sqrt{3}$ )

(D)

( $\dfrac{4}{\sqrt{3}}$ )

Q10. For which real values of $t$ are the vectors $\vec{v}_1=(1,t,t^2)$ , $\vec{v}_2=(t,1,t)$ and $\vec{v}_3=(t^2,t,1)$ linearly independent?

(A)

(All real $t$ )

(B)

( $t\neq 1$ )

(C)

( $t\in\mathbb{R}\setminus\{\pm 1\}$ )

(D)

( $t=\pm 1$ )

Q11. Que101: Let non-zero vectors $\mathbf{a}$ and $\mathbf{b}$ satisfy $|\mathbf{a}|=6$ , $|\mathbf{b}|=8$ and the scalar projection of $\mathbf{a}$ on $\mathbf{b}$ is $3$ . Find $|\mathbf{a}-\operatorname{proj}_{\mathbf b}\mathbf a|$ .

(A)

$3$

(B)

$4\sqrt{3}$

(C)

$3\sqrt{3}$

(D)

$3\sqrt{5}$

Q12. Que102: Find the shortest distance between the skew lines $\ell_1:\ \mathbf r=(1,0,1)+\lambda(1,1,1)$ and $\ell_2:\ \mathbf r=(2,1,0)+\mu(1,-1,1)$ .

(A)

$\sqrt{2}$

(B)

$\dfrac{4}{\sqrt{14}}$

(C)

$\dfrac{2}{\sqrt{7}}$

(D)

$0$

Q13. Que103: Let $\mathbf u=(1,1,0)$ and $\mathbf v=(1,-1,2)$ . Find the vector $\mathbf x$ of minimum magnitude satisfying $\mathbf x\cdot\mathbf u=3$ and $\mathbf x\cdot\mathbf v=1$ .

(A)

$\left(\tfrac{5}{3},\tfrac{4}{3},-\tfrac{1}{3}\right)$

(B)

$(2,1,0)$

(C)

$\left(\tfrac{3}{2},\tfrac{1}{6},\tfrac{1}{3}\right)$

(D)

$\left(\tfrac{5}{3},\tfrac{4}{3},\tfrac{1}{3}\right)$

Q14. Que104: Assertion (A): If non-zero vectors $\mathbf a,\mathbf b,\mathbf c$ satisfy $\mathbf a\times\mathbf b=\mathbf a\times\mathbf c$ , then $\mathbf b-\mathbf c$ is parallel to $\mathbf a$ .
Reason (R): For any vectors $\mathbf x,\mathbf y$ , $(\mathbf x\times\mathbf y)\cdot\mathbf x=0$ .

(A)

Both A and R are true and R is the correct explanation of A

(B)

Both A and R are true but R is not the correct explanation of A

(C)

A is true but R is false

(D)

A is false and R is true

Q15. Que105: For which real values of $k$ does the shortest distance between the lines $\ell_1:\ \mathbf r=(1,0,2)+t(1,1,1)$ and $\ell_2:\ \mathbf r=(2,k,0)+s(1,-1,2)$ equal $1$ ?

(A)

$7-\sqrt{14}$

(B)

$7+\sqrt{14}$

(C)

$7\pm\sqrt{14}$

(D)

No real value of $k$

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