N – 2.2 – Product of two vectors – Exercise Problems

Part – A

\[1.\ Show\ that\ the\ vectors\ \overrightarrow{i} – 3\overrightarrow{j} + 5\overrightarrow{k}\ and\ – 2\overrightarrow{i}+ 6\overrightarrow{j}+4\overrightarrow{k}\ are\ perpendicular\ to\ each\ other\ \hspace{10cm}\]
\[2.\ Find\ the\ value\ of\ p\ if\ the\ vectors\ 2\overrightarrow{i}+ \overrightarrow{j}- 5\overrightarrow{k}\ and\ p\overrightarrow{i}+ 3\overrightarrow{j} – 2\overrightarrow{k} are\ perpendicular.\ \hspace{10cm}\]
\[3.\ If\ |\overrightarrow{a}| = 3,\ |\overrightarrow{b}|= 5\ and\ |\overrightarrow{a} × \overrightarrow{b}|=10,\ find\ the\ angle\ between\ \overrightarrow{a}\ and\ \overrightarrow{b}.\ \hspace{10cm}\]

Part –B

\[1.\ Find\ the\ projection\ of\ the\ vector\ 2\overrightarrow{i}+ \overrightarrow{j}- 2\overrightarrow{k} on\ the\ vector\ \overrightarrow{i} – 2\overrightarrow{j} – 2\overrightarrow{k}\ \hspace{10cm}\]
\[2.\ Find\ the\ area\ of\ the\ parellelogram\ whose\ adjacent\ sides\ are\ \overrightarrow{i} + \overrightarrow{j} + 3\overrightarrow{k}\ and\ 2\overrightarrow{i} + \overrightarrow{j}+ 2\overrightarrow{k}.\ \hspace{10cm}\]

Part –C

\[1.\ Prove\ that\ the\ vectors\ \overrightarrow{i}+2\overrightarrow{j}+ \overrightarrow{k},\ \overrightarrow{i} + \overrightarrow{j}- 3\overrightarrow{k}\ and\ 7\overrightarrow{i}-4\overrightarrow{j}+\overrightarrow{k} are\ mutually\ perpendicular.\ \hspace{10cm}\]
\[2.\ Find\ the\ projection\ of\ the\ vector\ 8\overrightarrow{i}+ 4\overrightarrow{j}- 3\overrightarrow{k} on\ 2\overrightarrow{i} – 3 \overrightarrow{j} +2\overrightarrow{k}. Also\ find\ the\ angle\ between\ them \]
\[3.\ Find\ the\ area\ of\ the\ triangle\ formed\ by\ the\ points\ whose\ position\ vectors\ 5\overrightarrow{i}+2\overrightarrow{j} + 4\overrightarrow{k}\, \overrightarrow{i} +3\overrightarrow{j}+ 2\overrightarrow{k} and\ -\overrightarrow{i} – \overrightarrow{j}+\overrightarrow{k}.\]
\[4.\ Find\ the\ unit\ vector\ perpendicular\ to\ each\ of\ the\ vectors\ \overrightarrow{i} – \overrightarrow{j}+ 3\overrightarrow{k} and\ 2\overrightarrow{i}+ 3\overrightarrow{j} -\overrightarrow{k}.\ \hspace{10cm}\]\[Also\ find\ the\ sine\ of\ the\ angle\ between\ the\ vectors .\ \hspace{10cm}\]