In a right-angled triangle ABC; if the hypotenuse AB = p, then what is $\overrightarrow{\mathrm{AB}} . \overrightarrow{\mathrm{AC}} + \overrightarrow{\mathrm{BC}} . \overrightarrow{\mathrm{BA}} + \overrightarrow{\mathrm{CA}} . \overrightarrow{\mathrm{CB}} = ?$

• p

• p²

• 2p²

• $\frac{{\mathrm{p}}^2}{2}$

A force $\overrightarrow{\mathrm{F}} = 3\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - 4\hat{\mathrm{k}}$ is applied at the point(1, - 1, 2). What is the moment of the force about the point (2, - 1, 3) ?

• $\hat{\mathrm{i}} + 4\hat{\mathrm{j}} + 4\hat{\mathrm{k}}$

• $2\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}$

• $2\hat{\mathrm{i}} - 7\hat{\mathrm{j}} - 2\hat{\mathrm{k}}$

• $2\hat{\mathrm{i}} + 4\hat{\mathrm{j}} - \hat{\mathrm{k}}$

$\mathrm{If} \overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}} \mathrm{are} \mathrm{unit} \mathrm{vectors} \mathrm{and} \mathrm{\theta } \mathrm{is} \mathrm{the} \mathrm{angle} \mathrm{between} \mathrm{them}, \mathrm{then} \mathrm{what} \mathrm{is} {\sin }^2\left(\raisebox{1ex}{$\mathrm{\theta }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right) \mathrm{equal} \mathrm{to}$

• $\frac{{\left|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right|}^2}{4}$

• $\frac{{\left|\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}\right|}^2}{4}$

• $\frac{{\left|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right|}^2}{2}$

• $\frac{{\left|\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}\right|}^2}{2}$

Consider the following statements : 

1) The magnitude of $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}$ is same as the area of a triangle with sides $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$

2) If $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}} = \overrightarrow{0}$ where, $\overrightarrow{\mathrm{a}} \ne \overrightarrow{0}, \overrightarrow{\mathrm{b}} \ne \overrightarrow{0} ,\mathrm{then} \overrightarrow{\mathrm{a}} = \mathrm{\lambda }\overrightarrow{\mathrm{b}}$

Which of the above statements is/are correct ?

• 1 only

• 2 only

• Both 1 and 2

• Neither 1 nor 2

Consider the following equations for two vectors $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ :

1) $\left(\overline{\mathrm{a}} + \overline{\mathrm{b}}\right) . \left(\overline{\mathrm{a}} - \overline{\mathrm{b}}\right) = {\left|\overline{\mathrm{a}}\right|}^2 - {\left|\overline{\mathrm{b}}\right|}^2$

2) $\left(\left|\overline{\mathrm{a}} + \overline{\mathrm{b}}\right|\right) . \left(\left|\overline{\mathrm{a}} - \overline{\mathrm{b}}\right|\right) = {\left|\overline{\mathrm{a}}\right|}^2 - {\left|\overline{\mathrm{b}}\right|}^2$

3) ${\left|\overline{\mathrm{a}} \overline{\mathrm{b}}\right|}^2 + \left|\overline{\mathrm{a}} \times \overline{\mathrm{b}}\right| = {\left|\overline{\mathrm{a}}\right|}^2 {\left|\overline{\mathrm{b}}\right|}^2$

Which of the above statements are correct?

• 1, 2 and 3

• 1 and 2 only

• 1 and 3 only

• 2 and 3 only

If the magnitude of the sum of two non-zero vectors is equal to the magnitude of their difference, then which one of the following is correct ?

• The vectors are parallel

• The vectors are perpendicular

• The vectors are anti-parallel

• The vectors must be unit vectors

What is the scalar projection of $\overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} - 2\hat{\mathrm{j}} +\hspace{0.17em}\hat{\mathrm{k}} \mathrm{on} 4\hat{\mathrm{i}} - 4\hat{\mathrm{j}} + 7\hat{\mathrm{k}} ?$

• $\frac{\sqrt{6}}{9}$

• $\frac{19}{9}$

• $\frac{9}{19}$

• $\frac{\sqrt{6}}{19}$

48.

If the vectors $\overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}} = \hat{\mathrm{i}} +\hspace{0.17em}2\hat{\mathrm{j}} - 3\hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{c}} = \hat{\mathrm{j}} +\hspace{0.17em}\mathrm{p}\hat{\mathrm{k}}$ are coplanar, then what is the value of p ?

• 1

• - 1

• 5

• - 5

If $\left|\overline{\mathrm{a}} \times \overline{\mathrm{b}}\right| = 144 \mathrm{and} \left|\overline{\mathrm{a}}\right| = 4, \mathrm{then} \mathrm{what} \mathrm{is} \overline{\mathrm{b}} = ?$

• 3

• 4

• 6

• 8

Two adjacent sides of a parallelogram are  $2\hat{\mathrm{i}} - 4\hat{\mathrm{j}} + 5\hat{\mathrm{k}} \mathrm{and} \hat{\mathrm{i}} - 2\hat{\mathrm{j}} - 3\hat{\mathrm{k}}$ What is the magnitude of dot product of vectors which represent its diagonals?

• 21

• 25

• 31

• 36