Scalars and vectors; addition, subtraction and resolution — practice questions

Question 1 · difficulty L2 · understanding

The angle between vector $$\vec{Q}$$ and the resultant of $$(2 \vec{Q}+2 \vec{P})$$ and $$(2 \vec{Q}-2 \vec{P})$$ is :

• A. $$\tan ^{-1}(\mathrm{P} / \mathrm{Q})$$
• B. 0$$^\circ$$
• C. $$\tan ^{-1} \frac{(2 \vec{Q}-2 \vec{P})}{2 \vec{Q}+2 \vec{P}}$$
• D. $$\tan ^{-1}(2 Q / \mathrm{P})$$

Question 2 · difficulty L2 · understanding

If two vectors $$\vec{A}$$ and $$\vec{B}$$ having equal magnitude $$R$$ are inclined at angle $$\theta$$, then

• A. $$|\vec{A}+\vec{B}|=2 R \cos \left(\frac{\theta}{2}\right)$$
• B. $$|\vec{A}-\vec{B}|=2 R \cos \left(\frac{\theta}{2}\right)$$
• C. $$|\vec{A}-\vec{B}|=\sqrt{2} R \sin \left(\frac{\theta}{2}\right)$$
• D. $$|\vec{A}+\vec{B}|=2 R \sin \left(\frac{\theta}{2}\right)$$

Question 3 · difficulty L2 · understanding

A vector in $x-y$ plane makes an angle of $30^{\circ}$ with $y$-axis. The magnitude of $\mathrm{y}$-component of vector is $2 \sqrt{3}$. The magnitude of $x$-component of the vector will be :

• A. $\sqrt{3}$
• B. 2
• C. 6
• D. $\frac{1}{\sqrt{3}}$

Question 4 · difficulty L2 · understanding

When vector $$\vec{A}=2 \hat{i}+3 \hat{j}+2 \hat{k}$$ is subtracted from vector $$\overrightarrow{\mathrm{B}}$$, it gives a vector equal to $$2 \hat{j}$$. Then the magnitude of vector $$\overrightarrow{\mathrm{B}}$$ will be :

• A. 3
• B. $$\sqrt{33}$$
• C. $$\sqrt6$$
• D. $$\sqrt5$$

Question 5 · difficulty L2 · understanding

Two forces having magnitude $$A$$ and $$\frac{A}{2}$$ are perpendicular to each other. The magnitude of their resultant is:

• A. $$\frac{5 A}{2}$$
• B. $$\frac{\sqrt{5} A}{4}$$
• C. $$\frac{\sqrt{5} A}{2}$$
• D. $$\frac{\sqrt{5} A^{2}}{2}$$

Question 6 · difficulty L2 · understanding

Two vectors $$\overrightarrow A$$ and $$\overrightarrow B$$ have equal magnitudes. If magnitude of $$\overrightarrow A$$ + $$\overrightarrow B$$ is equal to two times the magnitude of $$\overrightarrow A$$ $$-$$ $$\overrightarrow B$$, then the angle between $$\overrightarrow A$$ and $$\overrightarrow B$$ will be :

• A. $${\sin ^{ - 1}}\left( {{3 \over 5}} \right)$$
• B. $${\sin ^{ - 1}}\left( {{1 \over 3}} \right)$$
• C. $${\cos ^{ - 1}}\left( {{3 \over 5}} \right)$$
• D. $${\cos ^{ - 1}}\left( {{1 \over 3}} \right)$$

Question 7 · difficulty L2 · understanding

Which of the following relations is true for two unit vector $$\widehat A$$ and $$\widehat B$$ making an angle $$\theta$$ to each other?

• A. $$|\widehat A + \widehat B| = |\widehat A - \widehat B|\tan {\theta \over 2}$$
• B. $$|\widehat A - \widehat B| = |\widehat A + \widehat B|\tan {\theta \over 2}$$
• C. $$|\widehat A + \widehat B| = |\widehat A - \widehat B|cos{\theta \over 2}$$
• D. $$|\widehat A - \widehat B| = |\widehat A + \widehat B|\cos {\theta \over 2}$$

Question 8 · difficulty L2 · understanding

The resultant of these forces $$\overrightarrow {OP} ,\overrightarrow {OQ} ,\overrightarrow {OR} ,\overrightarrow {OS}$$ and $$\overrightarrow {OT}$$ is approximately .......... N. [Take $$\sqrt 3 = 1.7$$, $$\sqrt 2 = 1.4$$ Given $$\widehat i$$ and $$\widehat j$$ unit vectors along x, y axis]

• A. $$9.25\widehat i + 5\widehat j$$
• B. $$3\widehat i + 15\widehat j$$
• C. $$2.5\widehat i - 14.5\widehat j$$
• D. $$- 1.5\widehat i - 15.5\widehat j$$

Question 9 · difficulty L2 · understanding

What will be the projection of vector $$\overrightarrow A = \widehat i + \widehat j + \widehat k$$ on vector $$\overrightarrow B = \widehat i + \widehat j$$ ?

• A. $$\sqrt 2 (\widehat i + \widehat j + \widehat k)$$
• B. $$(\widehat i + \widehat j)$$
• C. $$\sqrt 2 (\widehat i + \widehat j)$$
• D. $$2(\widehat i + \widehat j + \widehat k)$$

Question 10 · difficulty L2 · understanding

A vector has magnitude same as that of $$\vec{A}=3 \hat{i}+4 \hat{j}$$ and is parallel to $$\vec{B}=4 \hat{i}+3 \hat{j}$$. The $$x$$ and $$y$$ components of this vector in first quadrant are $$x$$ and 3 respectively where $$x=$$ _________.