Mathematics : Vector Algebra

Let OB = $\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$ and OA = $4\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$. The distance of the point B from the straight line passing through A and parallel to the vector $2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$ is

• $\frac{7\sqrt{5}}{9}$

• $\frac{5\sqrt{7}}{9}$

• $\frac{3\sqrt{5}}{7}$

• $\frac{9\sqrt{5}}{7}$

If a = $\mathrm{\lambda }\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$ and b = $2\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + \mathrm{\lambda }\hat{\mathrm{k}}$  are at right angle, then the value of $\left|\mathrm{a} + \mathrm{b}\right| - \left|\mathrm{a} - \mathrm{b}\right|$ is

• 2

• 1

• 0

• - 1

Let the position vectors of the points A, B and C be a, b and c, respectively. Let Q be the point of intersection of the medians of the $∆\mathrm{ABC}$. Then, QA + QB + QC is equal to

• $\frac{\mathrm{a} + \mathrm{b} + \mathrm{c}}{2}$

• 2a + b + c

• a + b + c

• 0

If $\mathrm{a} = 2\hat{\mathrm{i}} - \hat{\mathrm{j}} - \mathrm{m}\hat{\mathrm{k}}$ and $\mathrm{b} = \frac{4}{7}\hat{\mathrm{i}} - \frac{2}{7}\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$  are collinear, then the value of m is equal to

• - 7

• - 1

• 2

• 7

Let $\mathrm{a} = 2\hat{\mathrm{i}} + 5\hat{\mathrm{j}} - 7\hat{\mathrm{k}}$ and $\mathrm{b} = \hat{\mathrm{i}} + 3\hat{\mathrm{j}} - 5\hat{\mathrm{k}}$.  Then, (3a - 5b) . (4a $\times$ 5b) is equal to

• - 7

• 0

• - 13

• 1

If a + 2b - c =  0 and $\mathrm{a} \times \mathrm{b} + \mathrm{b} \times \mathrm{c} + \mathrm{c} \times \mathrm{a} = \mathrm{\lambda }\left(\mathrm{a} + \mathrm{b}\right)$, then the value of $\mathrm{\lambda }$ is equal to

• 5

• 4

• 2

• - 2

If a + b and a - b are perpendicular and b = $3\hat{\mathrm{i}} - 4\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$, then $\left|\mathrm{a}\right|$ is equal to

• $\sqrt{41}$

• $\sqrt{39}$

• $\sqrt{19}$

• $\sqrt{29}$

If the vectors $2\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 6\hat{\mathrm{k}}, 2\hat{\mathrm{i}} + \mathrm{\lambda }\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$ and $2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + \hat{\mathrm{k}}$ are coplanar, then the value of $\mathrm{\lambda }$ is

• - 10

• 1

• 0

• 2

Let a be a unit vector. If (x - a) . (x + a) = 12, then the magnitude of x is

• $\sqrt{8}$

• $\sqrt{9}$

• $\sqrt{10}$

• $\sqrt{13}$

A force of magnitude 5 unit acting along the vector $2\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + \hat{\mathrm{k}}$ displaces the point of application from (1, 2, 3) to (5, 3, 7). Then the work done is :

• 50/7 unit

• 50/3 unit

• 25/3

• 25/4