\displaystyle \textbf{Question 1: }\text{Can a vector have direction angles }45^{\circ},\ 60^{\circ},\ 120^{\circ}?
\displaystyle \text{Answer:}
\displaystyle \text{Let a vector make angles }\alpha=45^\circ,\;\beta=60^\circ,\;\gamma=120^\circ\text{ with }OX,OY,OZ\text{ respectively.}
\displaystyle \text{Let }l,m,n\text{ be the direction cosines of the vector.}
\displaystyle l=\cos45^\circ=\frac{1}{\sqrt{2}},\;m=\cos60^\circ=\frac{1}{2},\;n=\cos120^\circ=-\frac{1}{2}.
\displaystyle l^2+m^2+n^2=\frac{1}{2}+\frac{1}{4}+\frac{1}{4}=1.
\displaystyle \text{Since the direction cosines satisfy }l^2+m^2+n^2=1.
\displaystyle \text{Hence, a vector can have direction angles }45^\circ,60^\circ,120^\circ.

\displaystyle \textbf{Question 2: }\text{Prove that }1,1,1\text{ cannot be direction cosines of a straight line.}
\displaystyle \text{Answer:}
\displaystyle \text{Let }l=1,\;m=1,\;n=1\text{ be the direction cosines of a straight line.}
\displaystyle l^2+m^2+n^2=1^2+1^2+1^2=3.
\displaystyle \text{But the direction cosines must satisfy }l^2+m^2+n^2=1.
\displaystyle \text{Hence, }(1,1,1)\text{ cannot be the direction cosines of a straight line.}

\displaystyle \textbf{Question 3: }\text{A vector makes an angle of }\frac{\pi}{4}\text{ with each of }x\text{-axis and } \\ y\text{-axis. Find the angle made by it with the }z\text{-axis.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the vector }\overrightarrow{OP}\text{ make angles }\alpha=45^\circ,\;\beta=45^\circ.
\displaystyle \text{Suppose }\overrightarrow{OP}\text{ is inclined at angle }\gamma\text{ to }OZ.
\displaystyle \text{Let }l,m,n\text{ be the direction cosines of }\overrightarrow{OP}.
\displaystyle l=\cos\frac{\pi}{4}=\frac{1}{\sqrt{2}},\;m=\cos\frac{\pi}{4}=\frac{1}{\sqrt{2}},\;n=\cos\gamma.
\displaystyle l^2+m^2+n^2=1.
\displaystyle \Rightarrow \frac{1}{2}+\frac{1}{2}+n^2=1.
\displaystyle \Rightarrow n=0.
\displaystyle \Rightarrow \gamma=\frac{\pi}{2}.

\displaystyle \textbf{Question 4: }\text{A vector }\overrightarrow{r}\text{ is inclined at equal acute angles to }x,y,z\text{-axes. If }|\overrightarrow{r}|=6.
\displaystyle \text{Answer:}
\displaystyle l=m=n.
\displaystyle 3l^2=1\Rightarrow l=\frac{1}{\sqrt{3}}.
\displaystyle \overrightarrow{r}=6\left(\frac{1}{\sqrt{3}}\widehat{i}+\frac{1}{\sqrt{3}}\widehat{j}+\frac{1}{\sqrt{3}}\widehat{k}\right)=2\sqrt{3}(\widehat{i}+\widehat{j}+\widehat{k}).

\displaystyle \textbf{Question 5: }\text{If }|\overrightarrow{r}|=8,\ \alpha=45^\circ,\beta=60^\circ.
\displaystyle \text{Answer:}
\displaystyle l=\frac{1}{\sqrt{2}},\;m=\frac{1}{2},\;n=\frac{1}{2}.
\displaystyle \overrightarrow{r}=8\left(\frac{1}{\sqrt{2}}\widehat{i}+\frac{1}{2}\widehat{j}+\frac{1}{2}\widehat{k}\right)=4(\sqrt{2}\widehat{i}+\widehat{j}+\widehat{k}).

\displaystyle \textbf{Question 6: }\text{Find the direction cosines of the following vectors:}\\ \text{(i) }2\widehat{i}+2\widehat{j}-\widehat{k}\qquad \text{(ii) }6\widehat{i}-2\widehat{j}-3\widehat{k}\qquad \text{(iii) }3\widehat{i}-4\widehat{k}.
\displaystyle \text{Answer:}
\displaystyle \text{(i) }|\overrightarrow{a}|=\sqrt{4+4+1}=3
\displaystyle l=\frac{2}{3},\;m=\frac{2}{3},\;n=-\frac{1}{3}
\displaystyle \text{(ii) }|\overrightarrow{b}|=\sqrt{36+4+9}=7
\displaystyle l=\frac{6}{7},\;m=-\frac{2}{7},\;n=-\frac{3}{7}
\displaystyle \text{(iii) }|\overrightarrow{c}|=\sqrt{9+0+16}=5
\displaystyle l=\frac{3}{5},\;m=0,\;n=-\frac{4}{5}

\displaystyle \textbf{Question 7: }\text{Find the angles with the axes for:}\\ \text{(i) }\widehat{i}-\widehat{j}+\widehat{k}\qquad \text{(ii) }\widehat{j}-\widehat{k}\qquad \text{(iii) }4\widehat{i}+8\widehat{j}+\widehat{k}.
\displaystyle \text{Answer:}
\displaystyle \text{(i) }|\overrightarrow{r}|=\sqrt{3},\; \cos\alpha=\frac{1}{\sqrt{3}},\cos\beta=-\frac{1}{\sqrt{3}},\cos\gamma=\frac{1}{\sqrt{3}}
\displaystyle \text{(ii) }|\overrightarrow{r}|=\sqrt{2},\; \cos\alpha=0,\cos\beta=\frac{1}{\sqrt{2}},\cos\gamma=-\frac{1}{\sqrt{2}}
\displaystyle \text{(iii) }|\overrightarrow{r}|=\sqrt{81},\; \cos\alpha=\frac{4}{9},\cos\beta=\frac{8}{9},\cos\gamma=\frac{1}{9}

\displaystyle \textbf{Question 8: }\text{Show that }\widehat{i}+\widehat{j}+\widehat{k}\text{ is equally inclined to the axes.}
\displaystyle \text{Answer:}
\displaystyle |\overrightarrow{r}|=\sqrt{3},\; l=m=n=\frac{1}{\sqrt{3}}
\displaystyle \Rightarrow \alpha=\beta=\gamma

\displaystyle \textbf{Question 9: }\text{Show that the direction cosines are }\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}.
\displaystyle \text{Answer:}
\displaystyle l=m=n
\displaystyle 3l^2=1\Rightarrow l=\frac{1}{\sqrt{3}}

\displaystyle \textbf{Question 10: }\text{If unit }\overrightarrow{a}\text{ makes angles }\frac{\pi}{3},\frac{\pi}{4}\text{ with }\widehat{i},\widehat{j}.
\displaystyle \text{Answer:}
\displaystyle l=\frac{1}{2},\;m=\frac{1}{\sqrt{2}}
\displaystyle l^2+m^2+n^2=1\Rightarrow n=\frac{1}{2}
\displaystyle \overrightarrow{a}=\frac{1}{2}\widehat{i}+\frac{1}{\sqrt{2}}\widehat{j}+\frac{1}{2}\widehat{k}

\displaystyle \textbf{Question 11: }\text{Find }\overrightarrow{r},\ |\overrightarrow{r}|=3\sqrt{2},\ \beta=\frac{\pi}{4},\gamma=\frac{\pi}{2}.
\displaystyle \text{Answer:}
\displaystyle m=\frac{1}{\sqrt{2}},\;n=0
\displaystyle l^2+m^2+n^2=1\Rightarrow l=\frac{1}{\sqrt{2}}
\displaystyle \overrightarrow{r}=3(\widehat{i}+\widehat{j})

\displaystyle \textbf{Question 12: }\text{If }|\overrightarrow{r}|=2\sqrt{3}\text{ and equal angles with axes.}
\displaystyle \text{Answer:}
\displaystyle l=m=n=\frac{1}{\sqrt{3}}
\displaystyle \overrightarrow{r}=2(\widehat{i}+\widehat{j}+\widehat{k})

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