The length of the projection of the line segment joining the points (5, –1, 4) and (4, –1, 3) on the plane, x + y + z = 7 is:

• $\sqrt{\frac{2}{3}}$

• $\frac{2}{\sqrt{3}}$

• 2/3

• 1/3

For any vector x, where i, j, k have their usual meanings the value of ${\left(\mathrm{x} \times \hat{\mathrm{i}}\right)}^2 + {\left(\mathrm{x} \times \hat{\mathrm{j}}\right)}^2 + {\left(\mathrm{x} \times \hat{\mathrm{k}}\right)}^2 \mathrm{where} \hat{\mathrm{i}}, \hat{\mathrm{j}}, \hat{\mathrm{k}}$ have their usual meanings, is equal to

• ${\left|\mathrm{x}\right|}^2$

• 2${\left|\mathrm{x}\right|}^2$

• 3${\left|\mathrm{x}\right|}^2$

• 4${\left|\mathrm{x}\right|}^2$

If the sum of two unit vectors is a unit vector, then the magnitude of their difference is

• $\sqrt{2}$ units

• 2 units

• $\sqrt{3}$ units

• $\sqrt{5}$ units

For non-zero vectors a and b, if $\left|\mathrm{a} + \mathrm{b}\right| < \left|\mathrm{a} - \mathrm{b}\right|$,  then a and b are

• collinear

• perpendicular to each other

• inclined at an acute angle

• inclined at an obtuse angle

385.

Which of the following is not always true?

• ${\left|\mathrm{a} + \mathrm{b}\right|}^2 = {\left|\mathrm{a}\right|}^2 + {\left|\mathrm{b}\right|}^2$, if a and b are perpendicular to each other

• $\left|\mathrm{a} + \mathrm{\lambda b}\right| \ge \left|\mathrm{a}\right| \mathrm{for} \mathrm{all} \mathrm{\lambda } \in \mathrm{R}$,  if a and b are perpendicular to each other

• ${\left|\mathrm{a} + \mathrm{b}\right|}^2 +\hspace{0.17em}{\left|\mathrm{a} - \mathrm{b}\right|}^2 = 2\left({\left|\mathrm{a}\right|}^2 + {\left|\mathrm{b}\right|}^2\right)$

• $\left|\mathrm{a} + \mathrm{\lambda b}\right| \ge \left|\mathrm{a}\right| \mathrm{for} \mathrm{all} \mathrm{\lambda } \in \mathrm{R}$, if a is parallel to b

If the four points with position vectors $- 2\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}, \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}, \hat{\mathrm{j}} - \hat{\mathrm{k}} \mathrm{and} \mathrm{\lambda }\hat{\mathrm{j}} + \hat{\mathrm{k}}$ are coplanar, then $\mathrm{\lambda }$ is equal to

• 1

• 2

• - 1

• 0

The equation $\mathrm{z}\overline{\mathrm{z}} + \mathrm{a}\overline{\mathrm{z}} + \overline{\mathrm{a}}\mathrm{z} + \mathrm{b} = 0$, represents a circle, if

• ${\left|\mathrm{a}\right|}^2 = \mathrm{b}$

• ${\left|\mathrm{a}\right|}^2 > \mathrm{b}$

• ${\left|\mathrm{a}\right|}^2 <\hspace{0.17em}\mathrm{b}$

•  None of the above

If a, b, c are non-coplanar unit vectors such that

$\mathrm{a} \times \left(\mathrm{b} \times \mathrm{c}\right) = \frac{\mathrm{b} + \mathrm{c}}{\sqrt{2}}$, then the angle between a and b is

• $\frac{3\mathrm{\pi }}{4}$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

• $\mathrm{\pi }$

If a, b, c and a', b',c' form a reciprocal system of vectors, then a . a'+ b . b' + c · c' is equal to

• 0

• 1

• 2

• 3

If a, b, c are three non - zero vectors which are pairwise non-collinear. Also, a + 3b is collinear with c and b + 2c is collinear with a, then a + 3b + 6c is

• c

• 0

• a + c

• a