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Vector Algebra Set-5 MCQ Online Practice Test | Class 12 | Quiz Tweek

Vector Algebra Set-5

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

Vector Algebra forms the foundation for understanding 3D geometry in Class 12 and is heavily used in board and competitive exams. Mastery of dot product, cross product, scalar triple product, and vector equations helps solve problems on angles between vectors, projections, coplanarity, shortest distances between lines, and plane geometry efficiently and accurately.

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Q1. If $\vec{a}=2\hat{i}+3\hat{j}-6\hat{k}$ and $\vec{b}=\hat{i}-4\hat{j}+3\hat{k}$ , the scalar component of $\vec{a}$ along $\vec{b}$ is:

(A)

$-\dfrac{28}{13}$

(B)

$-\dfrac{14\sqrt{26}}{13}$

(C)

$\dfrac{28}{\sqrt{26}}$

(D)

$-\dfrac{14}{\sqrt{13}}$

Q2. If non-zero vectors $\vec{a},\vec{b},\vec{c}$ satisfy $\vec{a}+\vec{b}+\vec{c}=0$ and $|\vec{a}|=3,\;|\vec{b}|=4,\;|\vec{c}|=5$ , then the angle between $\vec{a}$ and $\vec{b}$ is:

(A)

$60^\circ$

(B)

$120^\circ$

(C)

$30^\circ$

(D)

$90^\circ$

Q3. Let $A(1,0,1)$ and $B(3,2,4)$ . The coordinates of the point on the line $AB$ closest to the origin are:

(A)

$\left(\dfrac{7}{17},-\dfrac{10}{17},\dfrac{2}{17}\right)$

(B)

$\left(\dfrac{1}{17},\dfrac{2}{17},-\dfrac{4}{17}\right)$

(C)

$\left(\dfrac{3}{17},-\dfrac{1}{17},\dfrac{5}{17}\right)$

(D)

$\left(\dfrac{2}{17},-\dfrac{10}{17},\dfrac{1}{17}\right)$

Q4. If non-zero vectors $\vec{u},\vec{v}$ satisfy $\vec{u}\times(\vec{u}\times\vec{v})=2\vec{u}-3\vec{v}$ , then the value of $\dfrac{|\vec{u}\cdot\vec{v}|}{|\vec{u}|^2}$ is:

(A)

$\dfrac{3}{2}$

(B)

$\dfrac{1}{\sqrt{3}}$

(C)

$\dfrac{2}{3}$

(D)

$0$

Q5. Does there exist a vector $\vec{x}$ such that $\vec{x}\times(\hat{i}+\hat{j}+\hat{k})=\hat{i}+2\hat{j}-\hat{k}$ and $\vec{x}\cdot(\hat{i}+\hat{j}+\hat{k})=0$ ?

(A)

Yes, exactly one such vector exists.

(B)

No such vector exists.

(C)

Yes, infinitely many such vectors exist.

(D)

Yes, but the number of such vectors is finite and greater than one.

Q6. Let $\mathbf{u},\mathbf{v}$ be unit vectors with $\mathbf{u}\cdot\mathbf{v}=\tfrac{1}{2}$ . The magnitude of $\mathbf{u}+2\mathbf{v}$ is

(A)

$\sqrt{5}$

(B)

$\sqrt{7}$

(C)

$3$

(D)

$\sqrt{3}$

Q7. Let $L$ be the line through $A(1,0,1)$ with direction vector $\mathbf{b}=(1,2,3)$ and let $P(2,1,-1)$ be a point. The shortest distance from $P$ to $L$ equals

(A)

$\dfrac{5}{\sqrt{14}}$

(B)

$\sqrt{\dfrac{19}{14}}$

(C)

$\sqrt{\dfrac{13}{7}}$

(D)

$\sqrt{\dfrac{75}{14}}$

Q8. For which values of $k$ are the vectors $\mathbf{a}=(1,2,k)$ , $\mathbf{b}=(2,3,4)$ and $\mathbf{c}=(3,k,6)$ coplanar?

(A)

$\{2,\tfrac{9}{2}\}$

(B)

$\{2\}$

(C)

$\{\tfrac{9}{2}\}$

(D)

$\{-1,\tfrac{9}{2}\}$

Q9. Let $\mathbf{u},\mathbf{v},\mathbf{w}$ be unit vectors in $\mathbb{R}^3$ satisfying $\mathbf{u}+\mathbf{v}+\mathbf{w}=0$ . The value of $\,\big|\mathbf{u}\times\mathbf{v}+\mathbf{v}\times\mathbf{w}+\mathbf{w}\times\mathbf{u}\big|\,$ is

(A)

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

(B)

$\dfrac{3}{2}$

(C)

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

(D)

$\dfrac{9}{4}$

Q10. The lines $L_1:\; \mathbf{r}=(1,2,3)+s(2,-1,1)$ and $L_2:\; \mathbf{r}=(3,0,1)+t(1,1,k)$ intersect for which value of $k$ ?

(A)

$3$

(B)

$5$

(C)

$7$

(D)

$-5$

(A)

$6$

(B)

$12$

(C)

$-12$

(D)

$24$

Q12. Find the value of $k$ for which the lines

\vec r=\begin{pmatrix}0\\0\\0\end{pmatrix}+\lambda\begin{pmatrix}1\\-1\\2\end{pmatrix} \quad\text{and}\quad \vec r=\begin{pmatrix}2\\0\\1\end{pmatrix}+\mu\begin{pmatrix}2\\1\\k\end{pmatrix}

are perpendicular and intersect.

(A)

$-1$

(B)

$0$

(C)

$\dfrac{1}{2}$

(D)

$-\dfrac{1}{2}$

Q13. Let $\vec a,\vec b\in\mathbb{R}^3$ satisfy $|\vec a|=1,\ |\vec b|=2,\ \vec a\cdot\vec b=1$ . If $\vec c$ is a unit vector perpendicular to $\vec a-\vec b$ , what is the maximum possible value of $(\vec a+\vec b)\cdot\vec c$ ?

(A)

$2$

(B)

$1$

(C)

$\sqrt{7}$

(D)

$0$

Q14. The shortest distance between the skew lines

\vec r=\begin{pmatrix}1\\0\\0\end{pmatrix}+\lambda\begin{pmatrix}1\\1\\1\end{pmatrix} \quad\text{and}\quad \vec r=\begin{pmatrix}0\\1\\2\end{pmatrix}+\mu\begin{pmatrix}2\\-1\\1\end{pmatrix}

(A)

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

(B)

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

(C)

$\dfrac{\sqrt{14}}{2}$

(D)

$\dfrac{7}{4}$

Q15. Let $\vec a,\vec b,\vec c$ be non-zero vectors in $\mathbb{R}^3$ . If $\vec a\times\vec b$ is parallel to $\vec b\times\vec c$ , then which of the following must be true?

(A)

$\vec a$ is parallel to $\vec c$ .

(B)

$\vec a,\vec b,\vec c$ are coplanar.

(C)

$\vec b\cdot(\vec a\times\vec c)\neq 0$ .

(D)

$\vec a,\vec b,\vec c$ are collinear.

...and 5 more challenging questions available in the interactive simulator.

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