Scalar and vector products — practice questions

Question 1 · difficulty L2 · understanding

Let $\overrightarrow{\mathrm{a}}=2 \hat{i}+3 \hat{j}+3 \hat{k}$ and $\overrightarrow{\mathrm{b}}=6 \hat{i}+3 \hat{j}+3 \hat{k}$. Then the square of the area of the triangle with adjacent sides determined by the vectors $(2 \vec{a}+3 \vec{b})$ and $(\vec{a}-\vec{b})$ is :

• A. 450
• B. 900
• C. 1800
• D. 2400

Question 2 · difficulty L2 · understanding

Let $$|\vec{a}|=2,|\vec{b}|=3$$ and the angle between the vectors $$\vec{a}$$ and $$\vec{b}$$ be $$\frac{\pi}{4}$$. Then $$|(\vec{a}+2 \vec{b}) \times(2 \vec{a}-3 \vec{b})|^{2}$$ is equal to :

• A. 441
• B. 482
• C. 841
• D. 882

Question 3 · difficulty L2 · understanding

Let $$\vec{a}=5 \hat{i}-\hat{j}-3 \hat{k}$$ and $$\vec{b}=\hat{i}+3 \hat{j}+5 \hat{k}$$ be two vectors. Then which one of the following statements is TRUE ?

• A. Projection of $$\vec{a}$$ on $$\vec{b}$$ is $$\frac{-13}{\sqrt{35}}$$ and the direction of the projection vector is opposite to the direction of $$\vec{b}$$.
• B. Projection of $$\vec{a}$$ on $$\vec{b}$$ is $$\frac{13}{\sqrt{35}}$$ and the direction of the projection vector is opposite to the direction of $$\vec{b}$$.
• C. Projection of $$\vec{a}$$ on $$\vec{b}$$ is $$\frac{13}{\sqrt{35}}$$ and the direction of the projection vector is same as of $$\vec{b}$$.
• D. Projection of $$\vec{a}$$ on $$\vec{b}$$ is $$\frac{-13}{\sqrt{35}}$$ and the direction of the projection vector is same as of $$\vec{b}$$.

Question 4 · difficulty L2 · understanding

Let $\vec{a}$ and $\vec{b}$ be two vectors, Let $|\vec{a}|=1,|\vec{b}|=4$ and $\vec{a} \cdot \vec{b}=2$. If $\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}$, then the value of $\vec{b} \cdot \vec{c}$ is :

• A. $-48$
• B. $-60$
• C. $-84$
• D. $-24$

Question 5 · difficulty L2 · understanding

Let $$\overrightarrow a = \widehat i + \widehat j - \widehat k$$ and $$\overrightarrow c = 2\widehat i - 3\widehat j + 2\widehat k$$. Then the number of vectors $$\overrightarrow b$$ such that $$\overrightarrow b \times \overrightarrow c = \overrightarrow a$$ and $$|\overrightarrow b | \in$$ {1, 2, ........, 10} is :

• A. 0
• B. 1
• C. 2
• D. 3

Question 6 · difficulty L2 · understanding

Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors such that $$|\vec{a}|=\sqrt{14},|\vec{b}|=\sqrt{6}$$ and $$|\vec{a} \times \vec{b}|=\sqrt{48}$$. Then $$(\vec{a} \cdot \vec{b})^{2}$$ is equal to ___________.

Question 7 · difficulty L3 · understanding

Let $\overrightarrow{\mathrm{a}}=4 \hat{i}-\hat{j}+3 \hat{k}, \overrightarrow{\mathrm{~b}}=10 \hat{i}+2 \hat{j}-\hat{k}$ and a vector $\overrightarrow{\mathrm{c}}$ be such that $2(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})+3(\overrightarrow{\mathrm{~b}} \times \overrightarrow{\mathrm{c}})=\overrightarrow{0}$. If $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}=15$, then $\overrightarrow{\mathrm{c}} \cdot(\hat{i}+\hat{j}-3 \hat{k})$ is equal to :

• A. -6
• B. -5
• C. -4
• D. -3

Question 8 · difficulty L3 · understanding

Let $\vec{a}=\sqrt{7} \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{j}+2 \hat{k}$. If $\vec{r}$ is a vector such that $\vec{r} \times \vec{a}+\vec{a} \times \vec{b}=\overrightarrow{0}$ and $\vec{r} \cdot \vec{a}=0$, then $|3 \vec{r}|^2$ is equal to:

• A. 44
• B. 54
• C. 86
• D. 132

Question 9 · difficulty L3 · understanding

Let $\hat{u}$ and $\hat{v}$ be unit vectors inclined at an acute angle such that $|\hat{u} \times \hat{v}|=\frac{\sqrt{3}}{2}$. If $\overrightarrow{\mathrm{A}}=\lambda \hat{u}+\hat{v}+(\hat{u} \times \hat{v})$, then $\lambda$ is equal to:

• A. $$\frac{4}{3}(\overrightarrow{\mathrm{~A}} \cdot \hat{u})-\frac{2}{3}(\overrightarrow{\mathrm{~A}} \cdot \hat{v})$$
• B. $$\frac{2}{3}(\overrightarrow{\mathrm{~A}} \cdot \hat{u})-\frac{1}{3}(\overrightarrow{\mathrm{~A}} \cdot \hat{v})$$
• C. $$\frac{4}{3}(\overrightarrow{\mathrm{~A}} \cdot \hat{u})+\frac{2}{3}(\overrightarrow{\mathrm{~A}} \cdot \hat{v})$$
• D. $$(\overrightarrow{\mathrm{A}} \cdot \hat{u})-\frac{1}{2}(\overrightarrow{\mathrm{~A}} \cdot \hat{v})$$

Question 10 · difficulty L3 · understanding

Let the vectors $\vec{a} = -\hat{i} + \hat{j} + 3\hat{k}$ and $\vec{b} = \hat{i} + 3\hat{j} + \hat{k}$. For some $\lambda, \mu \in \mathbb{R}$, let $\vec{c} = \lambda \vec{a} + \mu \vec{b}$. If $\vec{c} \cdot (3\hat{i} - 6\hat{j} + 2\hat{k}) = 10$ and $\vec{c} \cdot (\hat{i} + \hat{j} + \hat{k}) = -2$, then $|\vec{c}|^2$ is equal to :

• A. 8
• B. 12
• C. 14
• D. 15