Mark against the correct answer in each of the following:
Question: Mark $(\sqrt{ })$ against the correct answer in each of the following: If $\vec{a}, \vec{b}$ and $\vec{c}$ are mutually perpendicular unit vectors then $[\vec{a}+\vec{b}+\vec{c}]=$ ? A. 1 B. $\sqrt{2}$ C. $\sqrt{3}$ D. 2 Solution:...
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Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}$ are unit vectors such that $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{0}$ then $(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{a}}...
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Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: The unit vector normal to the plane containing $\overrightarrow{\mathrm{a}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{b}}=(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})$ is A. $(\hat{\mathrm{j}}-\hat{\mathrm{k}})$ B. $(-\hat{j}+\hat{k})$ C. $\frac{1}{\sqrt{2}}(-\hat{\mathrm{j}}+\hat{\mathrm{k}})$ D. $\frac{1}{\sqrt{2}}(-\hat{\mathrm{i}}+\hat{\mathrm{k}})$ Solution:...
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Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: Two adjacent sides of a triangle are represented by the vectors $\overrightarrow{\mathrm{a}}=3 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}$ and $\overrightarrow{\mathrm{b}}=-5 \hat{\mathrm{i}}+7 \hat{\mathrm{j}}$ The area of the triangle is A. 41 sq units B. 37 sq units C. $\frac{41}{2}$ sq units D. none of these Solution:...
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Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: The diagonals of a $\mathrm{II} \mathrm{gm}$ are represented by the vectors $\overrightarrow{\mathrm{d}_{1}}=(3 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{d}_{2}}=(\hat{\mathrm{i}}-3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})$ The area of the II gm is A. $7 \sqrt{3}$ sq units B. $5 \sqrt{3}$ sq units C. $3 \sqrt{5}$ sq units D. none of these Solution:...
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Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: Two adjacent sides of a $\| \mathrm{gm}$ are represented by the vectors $\vec{a}=(3 \hat{i}+\hat{j}+4 \hat{k})$ and $\overrightarrow{\mathrm{b}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})$ The area of the II gm is A. $\sqrt{42}$ sq units B. 6 sq units C. $\sqrt{35}$ sq units D. none of these Solution:...
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Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: If $|\overrightarrow{\mathrm{a}}|=\sqrt{26},|\overrightarrow{\mathrm{b}}|=7$ and $|\vec{a} \times \vec{b}|=35$ then$\vec{a} \cdot \vec{b}=?$ A. 5 B. 7 C. 13 D. 12 Solution:...
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Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: If $|\vec{a}|=2,|\vec{b}|=7$ and $(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})=(3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+6 \hat{\mathrm{k}})$ then the angle between $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ is A. $\frac{\pi}{6}$ B. $\frac{\pi}{3}$ C. $\frac{2 \pi}{3}$ D. $\frac{3 \pi}{4}$ Solution:...
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Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: If $\vec{a}=(\hat{i}-2 \hat{j}+3 \hat{k})$ and $\overrightarrow{\mathrm{b}}=(\hat{\mathrm{i}}-3 \hat{\mathrm{k}})$ then $|\overrightarrow{\mathrm{b}} \times 2 \overrightarrow{\mathrm{a}}|=?$ A. $10 \sqrt{3}$ B. $5 \sqrt{17}$ c. $4 \sqrt{19}$ D. $2 \sqrt{23}$ Solution:...
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Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: If $\overrightarrow{\mathrm{a}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{b}}=(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-4 \hat{\mathrm{k}})$ then $|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|=$ ? A. $\sqrt{174}$ B. $\sqrt{87}$ C. $\sqrt{93}$ D. none of these Solution:...
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Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: If $\boldsymbol{\theta}$ is the angle between two unit vectors $\hat{\mathrm{a}}$ and $\hat{\mathrm{b}}$ then $\frac{1}{2}|\hat{\mathrm{a}}-\hat{\mathrm{b}}|=$ ? A. $\cos \frac{\theta}{2}$ B. $\sin \frac{\theta}{2}$ C. $\tan \frac{\theta}{2}$ D. none of these Solution:...
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Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: If the vectors $\overrightarrow{\mathrm{a}}=3 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+\lambda \hat{\mathrm{j}}-3 \hat{\mathrm{k}}$ are perpendicular to each other then $\boldsymbol{\lambda}=$ ? A. $-3$ B. $-6$ C. $-9$ D. $-1$ Solution:...
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Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: If $\vec{a}$ and $\vec{b}$ are mutually perpendicular unit vectors then $(3 \vec{a}+2 \vec{b}) \cdot(5 \vec{a}-6 \vec{b})=?$ A. 3 B. 5 C. 6 D. 12 Solution:...
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Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: If $|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|=|\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|$ then A. $|\vec{a}|=|\vec{b}|$ B. $\vec{a} \| \vec{b}$ C. $\vec{a} \perp \vec{b}$ D. none of these Solution:...
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Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: What is the projection of $\vec{a}=(2 \hat{j}-\hat{j}+\hat{k})$ on $\overrightarrow{\mathrm{b}}=(\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}})$ $?$ A. $\frac{2}{\sqrt{3}}$ B. $\frac{4}{\sqrt{5}}$ C. $\frac{5}{\sqrt{6}}$ D. none of these Solution:...
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Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: If $\overrightarrow{\mathrm{a}}=(2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-\hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{b}}=(3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\lambda \hat{\mathrm{k}})$ be such that $\vec{a} \perp \vec{b}$ then $\boldsymbol{\lambda}=$ ? A. 2 B. $-2$ C. 3 D. $-3$ Solution:...
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Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: If $\vec{a}=(\hat{i}+2 \hat{j}-3 \hat{k})$ and $\overrightarrow{\mathrm{b}}=(3 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})$ then the angle between $(2 \vec{a}+\vec{b})$ and $(\vec{a}+2 \vec{b})$ is A. $\cos ^{-1}\left(\frac{21}{40}\right)$ B. $\cos ^{-1}\left(\frac{31}{50}\right)$ C. $\cos ^{-1}\left(\frac{11}{30}\right)$ D. none of these Solution:...
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Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: If $\overrightarrow{\mathrm{a}}=(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{b}}=(3 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})$ then the angle between $(\vec{a}+\vec{b})$ and $(\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}})$ is A. $\frac{\pi}{3}$ B. $\frac{\pi}{4}$ C. $\frac{\pi}{2}$ D. $\frac{2 \pi}{3}$ Solution:...
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Question: Mark ( $\sqrt{)}$ against the correct answer in each of the following: The angle between the vectors $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ is A. $\cos ^{-1} \frac{5}{7}$ B. $\cos ^{-1} \frac{3}{5}$ C. $\cos ^{-1} \frac{3}{\sqrt{14}}$ D. none of these Solution:...
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Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: If $\vec{a}$ and $\vec{b}$ are two vectors such that $|\vec{a}|=|\vec{b}|=\sqrt{2}$ and $\vec{a} \cdot \vec{b}=-1$ then the angle between $\vec{a}$ and $\vec{b}$ is A. $\frac{\pi}{6}$ B. $\frac{\pi}{4}$ C. $\frac{\pi}{3}$ D. $\frac{2 \pi}{3}$ Solution:...
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Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: f $\vec{a}$ and $\vec{b}$ are vectors such that $|\vec{a}|=\sqrt{3},|\vec{b}|=2$ and $\vec{a} \cdot \vec{b}=\sqrt{6}$ then the angle between $\vec{a}$ and $\overrightarrow{\mathrm{b}}$ is A. $\frac{\pi}{6}$ B. $\frac{\pi}{3}$ C. $\frac{\pi}{4}$ D. $\frac{2 \pi}{3}$ Solution:...
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Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: What is the angle which the vector $(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\sqrt{2} \hat{\mathrm{k}})$ makes with the z-axis? A. $\frac{\pi}{4}$ B. $\frac{\pi}{3}$ C. $\frac{\pi}{6}$ D. $\frac{2 \pi}{3}$ Solution:...
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Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: The vector $(\cos \alpha \cos \beta) \hat{i}+(\cos \alpha \cos \beta) \hat{j}+(\sin \alpha) \hat{k}$ is a A. null vector B. unit vector C. a constant vector D. none of these Solution:...
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Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: If a vector makes angle $a, \beta$ and $y$ with the $x$-axis, $y$-axis and $z$-axis respectively then the value of $\left(\sin ^{2} a+\sin ^{2} \beta+\sin ^{2} y\right)$ is A. 1 B. 2 C. 0 D. 3 Solution:...
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Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: If $A(1,2,-3)$ and $B(-1,-2,1)$ are the end points of a vector $\overrightarrow{A B}$ then the direction cosines of $\overrightarrow{\mathrm{AB}}$ are A. $-2,-4,4$ B. $\frac{-1}{2},-1,1$ C. $\frac{-1}{3}, \frac{-2}{3}, \frac{2}{3}$ D. none of these Solution:...
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