N – 2.2 – Product of two vectors – Exercise Problems

\[\underline{PART\ -\ A}\]
\[1.\ What\ are\ the\ values\ of\ \overrightarrow{i}\ . \ \overrightarrow{j}\ ,\ and\ \overrightarrow{k}\ .\ \overrightarrow{k}\ \hspace{18cm}\]
\[2.\ Find\ the\ scalar\ product\ of\ \overrightarrow{i}+\overrightarrow{j}\ ,\ \overrightarrow{i}+\overrightarrow{j}+ 3\overrightarrow{k}\ \hspace{10cm}\]
\[3.\ Show\ that\ the\ vectors\ \overrightarrow{i} – 3\overrightarrow{j} + 5\overrightarrow{k}\ and\ – 2\overrightarrow{i}+ 6\overrightarrow{j}+4\overrightarrow{k}\ are\ perpendicular\ to\ each\ other\ \hspace{10cm}\]
\[4.\ Find\ the\ value\ of\ p\ if\ the\ vectors\ 2\overrightarrow{i}+ \overrightarrow{j}- 5\overrightarrow{k}\ and\ p\overrightarrow{i}+ 3\overrightarrow{j} – 2\overrightarrow{k} are\ perpendicular.\ \hspace{10cm}\]
\[5.\ If\ \overrightarrow{a}\ and\ \overrightarrow{b}\ are\ two\ vectors\ such\ that\ |\overrightarrow{a}| = 6,\ |\overrightarrow{b}|= 4\ and\ \overrightarrow{a}\ . \overrightarrow{b}\ =12\ \hspace{5cm}\]\[find\ the\ angle\ between\ them.\ \hspace{3cm}\]
\[6.\ What\ are\ the\ values\ of\ (i)\ \overrightarrow{i}\ . \ \overrightarrow{i}\ ,\ and\ (ii)\ \overrightarrow{i}\ ×\ \overrightarrow{j}\ \hspace{18cm}\]
\[7.\ Prove\ that\ the\ vectors\ \overrightarrow{a}\ =\ 4\overrightarrow{i}\ -\ 2\overrightarrow{j}\ -\ 6\overrightarrow{k}\ and\ \overrightarrow{b}\ =\ 2\overrightarrow{i}\ -\ \overrightarrow{j}\ -\ 3\overrightarrow{k}\ are\ parallel.\ \hspace{10cm}\]
\[8.\ If\ \overrightarrow{a}\ and\ \overrightarrow{b}\ are\ two\ adjacent\ sides\ of\ a\ parallelogram.\ \hspace{15cm}\]\[What\ is\ its\ area?\ \hspace{12cm}\]
\[9.\ If\ |\overrightarrow{a}| = 3,\ |\overrightarrow{b}|= 5\ and\ |\overrightarrow{a} × \overrightarrow{b}|=10,\ find\ the\ angle\ between\ \overrightarrow{a}\ and\ \overrightarrow{b}.\ \hspace{10cm}\]
\[\underline{PART\ -\ B}\]
\[10.\ Find\ the\ projection\ of\ the\ vector\ 2\overrightarrow{i}+ \overrightarrow{j}- 2\overrightarrow{k} on\ the\ vector\ \overrightarrow{i} – 2\overrightarrow{j} – 2\overrightarrow{k}\ \hspace{10cm}\]
\[11.\ Find\ the\ projection\ of\ the\ vector\ 2\overrightarrow{i}\ +\ 3\overrightarrow{j}\ +\ \overrightarrow{k}\ on\ the\ vector\ 3\overrightarrow{i}\ -\ \overrightarrow{j}\ +\ \overrightarrow{k}\ \hspace{10cm}\]
\[12.\ Find\ the\ area\ of\ the\ parellelogram\ whose\ adjacent\ sides\ are\ \overrightarrow{i} + \overrightarrow{j} + 3\overrightarrow{k}\ and\ 2\overrightarrow{i} + \overrightarrow{j}+ 2\overrightarrow{k}.\ \hspace{10cm}\]
\[13.\ If\ \overrightarrow{d_1}\ =\ 4\overrightarrow{i}\ +\ 2\overrightarrow{j}\ +\ 3\overrightarrow{k}\ and\ \overrightarrow{d_2}\ =\ \overrightarrow{i}\ -\ \overrightarrow{j}\ +\ \overrightarrow{k}\ are\ \hspace{15cm}\]\[diagonals\ of\ a\ parellelogram.\ Find\ its\ Area\ \hspace{10cm}\]
\[14.\ Find\ the\ area\ of\ the\ triangle\ whose\ adjacent\ sides\ are\ 2\overrightarrow{i}\ +\ 3\overrightarrow{j}\ -\ \overrightarrow{k} and\ \overrightarrow{i}\ +\ 3\overrightarrow{j}\ +\ \overrightarrow{k}.\]
\[\underline{PART\ -\ C}\]
\[15.\ If\ \overrightarrow{a}\ = \ 2\overrightarrow{i}\ +\ \overrightarrow{j}\ +\ 3\overrightarrow{k}\ and\ \overrightarrow{b}\ =\ \overrightarrow{i} -\ 4\overrightarrow{j}\ -\ 6\overrightarrow{k},\ \hspace{15cm}\]\[find\ the\ projection\ of\ \overrightarrow{a}\ on\ \overrightarrow{b}\ .\ Also\ find\ the\ angle\ between\ them\ \hspace{5cm}\]
\[16.\ Prove\ that\ the\ vectors\ \overrightarrow{i}+2\overrightarrow{j}+ \overrightarrow{k},\ \overrightarrow{i} + \overrightarrow{j}- 3\overrightarrow{k}\ and\ 7\overrightarrow{i}-4\overrightarrow{j}+\overrightarrow{k} are\ mutually\ perpendicular.\ \hspace{10cm}\]
\[17.\ Show\ that\ (\overrightarrow{a}\ .\ \overrightarrow{i})\overrightarrow{i}\ +\ (\overrightarrow{a}\ .\ \overrightarrow{j})\overrightarrow{j}\ +\ (\overrightarrow{a}\ .\ \overrightarrow{k})\overrightarrow{k}\ =\ \overrightarrow{a}\ ,\ if\ \overrightarrow{a}\ is\ any\ vector\ \hspace{10cm}\]
\[18.\ Find\ the\ area\ of\ the\ triangle\ formed\ by\ the\ points\ whose\ position\ vectors\ \hspace{15cm}\]\[5\overrightarrow{i}+2\overrightarrow{j} + 4\overrightarrow{k}\ ,\ \overrightarrow{i} +3\overrightarrow{j}+ 2\overrightarrow{k} and\ -\overrightarrow{i} – \overrightarrow{j}+\overrightarrow{k}\ \hspace{5cm}\]
\[19.\ Find\ the\ unit\ vector\ perpendicular\ to\ each\ of\ the\ vectors\ \hspace{15cm}\]\[3\overrightarrow{i}\ +\ 3\overrightarrow{j}+ \overrightarrow{k}\ and\ 2\overrightarrow{i}\ -\ 5\overrightarrow{j}\ +\ 3\overrightarrow{k}.\ \hspace{5cm}\]
\[20.\ Find\ the\ unit\ vector\ perpendicular\ to\ each\ of\ the\ vectors\ \overrightarrow{i} – \overrightarrow{j}+ 3\overrightarrow{k} and\ 2\overrightarrow{i}+ 3\overrightarrow{j} -\overrightarrow{k}.\ \hspace{10cm}\]\[Also\ find\ the\ sine\ of\ the\ angle\ between\ the\ vectors .\ \hspace{10cm}\]
\[21.\ The\ forces\ 2\overrightarrow{i}\ -\ 5\overrightarrow{j}\ + \ 6\overrightarrow{k}\ ,\ -\overrightarrow{i}\ +\ 2\overrightarrow{j}\ – \ \overrightarrow{k}\ and\ 2\overrightarrow{i}\ +\ 7\overrightarrow{j}\ act\ on\ a\ particle\ \hspace{15cm}\]\[and\ displace\ it\ from\ the\ point\ 4\overrightarrow{i}\ -\ 3\overrightarrow{j}\ -\ 2\overrightarrow{k}\ to\ the\ point\ 6\overrightarrow{i}\ +\ \overrightarrow{j}\ – \ 3\overrightarrow{k}.\ Find\ the\ total\ work\ done\ by\ the\ forces\ \hspace{8cm}\]
\[22.\ The\ forces\ 3\overrightarrow{i}\ +\ 5\overrightarrow{j}\ – \ 2\overrightarrow{k}\ and\ 2\overrightarrow{i}\ +\ 3\overrightarrow{j}\ – \ 5\overrightarrow{k}\ displaces\ a\ particle\ \hspace{15cm}\]\[from\ the\ point\ (1, 2, -1)\ to\ the\ point\ (5, -3, 4).\ Find\ the\ total\ work\ done\ by\ the\ force.\ \hspace{8cm}\]
\[23.\ Find\ the\ moment\ of\ the\ force\ 6\overrightarrow{i}\ +\ \overrightarrow{j}\ +\ \overrightarrow{k}\ acting\ through\ the\ point\ \hspace{10cm}\]\[\overrightarrow{i}\ +\ 2\overrightarrow{j}\ +\ 3\overrightarrow{k} about\ the\ point\ -\overrightarrow{i}\ – \overrightarrow{j}\ +\ \overrightarrow{k}.\ \hspace{10cm}\]
\[24.\ Find\ the\ magnitude\ of\ the\ torque\ about\ the\ point\ (4, 3, -1)\ of\ the\ force\ represented\ by\ 6\overrightarrow{i}\ +\ \overrightarrow{j}\ -\ \overrightarrow{k}\ \hspace{10cm}\]\[acting\ through\ the\ point\ (0, 1, -1)\ \hspace{10cm}\]