Scalar and vector products — practice questions

Question 1 · difficulty L2 · understanding

If two vectors $$\overrightarrow P = \widehat i + 2m\widehat j + m\widehat k$$ and $$\overrightarrow Q = 4\widehat i - 2\widehat j + m\widehat k$$ are perpendicular to each other. Then, the value of m will be :

• A. $$-1$$
• B. 3
• C. 1
• D. 2

Question 2 · difficulty L2 · understanding

$$\overrightarrow A$$ is a vector quantity such that $$|\overrightarrow A |$$ = non-zero constant. Which of the following expression is true for $$\overrightarrow A$$ ?

• A. $$\overrightarrow A \,.\,\overrightarrow A = 0$$
• B. $$\overrightarrow A \times \overrightarrow A < 0$$
• C. $$\overrightarrow A \times \overrightarrow A = 0$$
• D. $$\overrightarrow A \times \overrightarrow A > 0$$

Question 3 · difficulty L2 · understanding

If $$\overrightarrow A$$ and $$\overrightarrow B$$ are two vectors satisfying the relation $$\overrightarrow A$$ . $$\overrightarrow B$$ = $$\left| {\overrightarrow A \times \overrightarrow B } \right|$$. Then the value of $$\left| {\overrightarrow A - \overrightarrow B } \right|$$ will be :

• A. $$\sqrt {{A^2} + {B^2} + \sqrt 2 AB}$$
• B. $$\sqrt {{A^2} + {B^2}}$$
• C. $$\sqrt {{A^2} + {B^2} - \sqrt 2 AB}$$
• D. $$\sqrt {{A^2} + {B^2} + 2AB}$$

Question 4 · difficulty L2 · understanding

For three vectors $$\vec{A}=(-x \hat{i}-6 \hat{j}-2 \hat{k}), \vec{B}=(-\hat{i}+4 \hat{j}+3 \hat{k})$$ and $$\vec{C}=(-8 \hat{i}-\hat{j}+3 \hat{k})$$, if $$\vec{A} \cdot(\vec{B} \times \vec{C})=0$$, then value of $$x$$ is ________.

Question 5 · difficulty L2 · understanding

Vectors $$a\widehat i + b\widehat j + \widehat k$$ and $$2\widehat i - 3\widehat j + 4\widehat k$$ are perpendicular to each other when $$3a + 2b = 7$$, the ratio of $$a$$ to $$b$$ is $${x \over 2}$$. The value of $$x$$ is ____________.

Question 6 · difficulty L2 · understanding

If $$\overrightarrow P \times \overrightarrow Q = \overrightarrow Q \times \overrightarrow P$$, the angle between $$\overrightarrow P$$ and $$\overrightarrow Q$$ is $$\theta$$(0$$^\circ$$ < $$\theta$$ < 360$$^\circ$$). The value of '$$\theta$$' will be ___________$$^\circ$$.

Question 7 · difficulty L3 · understanding

Two particles are located at equal distance from origin. The position vectors of those are represented by $\vec{A}=2 \hat{i}+3 n \hat{j}+2 \hat{k}$ and $\bar{B}=2 \hat{i}-2 \hat{j}+4 p \hat{k}$, respectively. If both the vectors are at right angle to each other, the value of $n^{-1}$ is ________ .

Question 8 · difficulty L3 · understanding

The resultant of two vectors $$\vec{A}$$ and $$\vec{B}$$ is perpendicular to $$\vec{A}$$ and its magnitude is half that of $$\vec{B}$$. The angle between vectors $$\vec{A}$$ and $$\vec{B}$$ is _________$$^\circ$$.

Question 9 · difficulty L3 · understanding

If $$\vec{a}$$ and $$\vec{b}$$ makes an angle $$\cos ^{-1}\left(\frac{5}{9}\right)$$ with each other, then $$|\vec{a}+\vec{b}|=\sqrt{2}|\vec{a}-\vec{b}|$$ for $$|\vec{a}|=n|\vec{b}|$$ The integer value of $$\mathrm{n}$$ is _________.

Question 10 · difficulty L3 · understanding

Three particles P, Q and R are moving along the vectors $$\overrightarrow A = \widehat i + \widehat j$$, $$\overrightarrow B = \widehat j + \widehat k$$ and $$\overrightarrow C = - \widehat i + \widehat j$$ respectively. They strike on a point and start to move in different directions. Now particle P is moving normal to the plane which contains vector $$\overrightarrow A$$ and $$\overrightarrow B$$. Similarly particle Q is moving normal to the plane which contains vector $$\overrightarrow A$$ and $$\overrightarrow C$$. The angle between the direction of motion of P and Q is $${\cos ^{ - 1}}\left( {{1 \over {\sqrt x }}} \right)$$. Then the value of x is _______________.