Mathematics : Vector Algebra

For what value of $\mathrm{\lambda }$ are the vectors  $\overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} + \mathrm{\lambda } \hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}} = \hat{\mathrm{i}} - 2\hat{\mathrm{j}} -3\hat{\mathrm{k}}$ are perpendicular to each other?

$$\begin{gathered} \mathrm{If} \overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}} = \hat{\mathrm{j}} - \hat{\mathrm{k}}, \mathrm{find} \mathrm{a} \mathrm{vector} \overrightarrow{\mathrm{c}} \mathrm{such} \mathrm{that} \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}} = \overrightarrow{\mathrm{b}} \\ \mathrm{and} \overrightarrow{\mathrm{a}}.\overrightarrow{\mathrm{c}} = 3 \end{gathered}$$

If $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} = 0 \mathrm{and} \left|\overrightarrow{\mathrm{a}}\right| = 3, \left|\overrightarrow{\mathrm{b}}\right| = 5 \mathrm{and} \left|\overrightarrow{\mathrm{c}}\right| = 7,$ show that the angle between $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{b}$ is 60⁰.

Find the projection of $\overrightarrow{\mathrm{a}}$ on $\overrightarrow{\mathrm{b}}$ if $\overrightarrow{\mathrm{a}}. \overrightarrow{\mathrm{b}} =8 \mathrm{and} \overrightarrow{\mathrm{b}} = 2\hat{\mathrm{i}} + 6\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$

Write a unit vector in the direction of $\overrightarrow{\mathrm{b}} = 2\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}.$

Write the value of p for which $\overrightarrow{\mathrm{a}} = 3\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 9\hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}} = \hat{\mathrm{i}}+\mathrm{p}\hat{\mathrm{j}} +3\hat{\mathrm{k}}$ are parallel vectors.

$$\begin{gathered} \mathrm{If} \overrightarrow{\mathrm{a}} \mathrm{x} \overrightarrow{\mathrm{b}} = \overrightarrow{\mathrm{c}} \mathrm{x} \overrightarrow{\mathrm{d}} \mathrm{and} \overrightarrow{\mathrm{a}} \mathrm{x} \overrightarrow{\mathrm{c}} = \overrightarrow{\mathrm{b}} \mathrm{x} \overrightarrow{\mathrm{d}}, \mathrm{show} \mathrm{that} \overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{d}} \mathrm{is} \mathrm{parallel} \mathrm{to} \overrightarrow{\mathrm{b}} - \overrightarrow{\mathrm{c}}, \\ \mathrm{where} \overrightarrow{\mathrm{a}} \ne \overrightarrow{\mathrm{d}} \mathrm{and} \overrightarrow{\mathrm{b}} \ne \overrightarrow{\mathrm{c}} \end{gathered}$$

Write a vector of magnitude 15 units in the direction of vector  $\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 2\mathrm{k}$

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are  $\left( 2\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} \right) \mathrm{and} \left( \overrightarrow{\mathrm{a}} - 3\overrightarrow{\mathrm{b}} \right)$ respectively, externally in the ratio 1:2. Also, show that P is the midpoint of the line segment R.

For what value of ‘a’ the vectors   $2 \hat{\mathrm{i}} - 3 \hat{\mathrm{j}} + 4 \mathrm{k} \mathrm{and} \mathrm{a} \hat{\mathrm{i}} + 6 \hat{\mathrm{j}} - 8 \mathrm{k}$  are collinear?