2.1 – N – Vector Introduction – Exercise Problems

\[\underline{PART\ -\ A}\]
\[1.\ If\ \overrightarrow{a}= 3\overrightarrow{i}\ + 2\overrightarrow{j} + \overrightarrow{k}\ and\ \overrightarrow{b}= \overrightarrow{i}\ + 3\overrightarrow{j} + \overrightarrow{k}\, find\ 3\overrightarrow{a}\ +\ \overrightarrow{b}\ \hspace{10cm}\]
\[2.\ If\ \overrightarrow{a}= 2\overrightarrow{i}\ + 3\overrightarrow{j} + \overrightarrow{k}\ and\ \overrightarrow{b}= 3\overrightarrow{i}\ – \overrightarrow{j} + \overrightarrow{k}\, find\ 2\overrightarrow{a}\ +\ 3\overrightarrow{b}\ \hspace{20cm}\]
\[3.\ If\ position\ vectors\ of\ the\ points\ A\ and\ B\ are\ 2\overrightarrow{i}\ -\overrightarrow{j} + 3\overrightarrow{k}\ and\ 5\overrightarrow{i}\ + \overrightarrow{j} – 2\overrightarrow{k}\ find\ \overrightarrow{|AB|}\ \hspace{20cm}\]
\[4.\ Find\ the\ unit\ vector\ along\ the\ vector\ 2\overrightarrow{i}\ – \overrightarrow{j}- \overrightarrow{k}\ \hspace{20cm}\]
\[5.\ Find\ the\ direction\ cosines\ of\ the\ vector\ 3\overrightarrow{i}\ +\ 4 \overrightarrow{j}-\ 5 \overrightarrow{k}\ \hspace{20cm}\]
\[\underline{PART\ -\ B}\]
\[5.\ If\ the\ position\ vector\ of\ the\ points\ A\ and\ B\ are\ \hspace{15cm}\]\[\overrightarrow{i}\ -\ \overrightarrow{j}\ +\ \overrightarrow{k}\ and\ 3\overrightarrow{i}\ +\ 2\overrightarrow{j}\ +\ 3\overrightarrow{k},\ \hspace{12cm}\]\[find\ \overrightarrow{|AB|}\ ,\ Also\ find\ the\ direction\ ratio\ of\ \overrightarrow{AB}\ \hspace{10cm}\]
\[6.\ Find\ the\ Modulus\ and\ Direction\ cosines\ \hspace{15cm}\]\[of\ the\ vector\ 2\overrightarrow{i}\ + 3\overrightarrow{j}\ +\ 4\overrightarrow{k}\ \hspace{12cm}\]

7. Show that the points whose position vectors

\[2\overrightarrow{i}\ – \overrightarrow{j} + 3\overrightarrow{k},\ 3\overrightarrow{i}\ – 5\overrightarrow{j} + \overrightarrow{k}\ and\ -\overrightarrow{i}\ +11 \overrightarrow{j}+ 9\overrightarrow{k}\ are\ collinear\ \hspace{10cm}\]
\[\underline{PART\ -\ C}\]

8. Prove that the points

\[2\overrightarrow{i}\ + 3\overrightarrow{j}+ 4\overrightarrow{k}, 3\overrightarrow{i}\ + 4\overrightarrow{j}+ 2\overrightarrow{k} and\ 4\overrightarrow{i}\ +2 \overrightarrow{j}+ 3\overrightarrow{k}\ form\ an\ equilateral\ triangle\]
\[9.\ Prove\ that\ the\ points\ \hspace{15cm}\]
\[3\overrightarrow{i}\ -\ \overrightarrow{j}\ -\ 2 \overrightarrow{k}\ ,\ 5\overrightarrow{i}\ +\ \overrightarrow{j}\ -\ 3\overrightarrow{k} and\ 6\overrightarrow{i}\ -\ \overrightarrow{j}\ -\ \overrightarrow{k}\ form\ an\ isosceles\ triangle\]
\[10.\ The\ position\ vectors\ of\ the\ \triangle\ ABC\ are\ \hspace{18cm}\]\[2\overrightarrow{i}\ – \overrightarrow{j}+ \overrightarrow{k}, \overrightarrow{i}\ – 3\overrightarrow{j} – 5\overrightarrow{k} and\ 3\overrightarrow{i}\ -4 \overrightarrow{j} – 4\overrightarrow{k}\ respectively.\ Prove\ that\ triangle\ is\ right\ angled\ \hspace{10cm}\]

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