Vectors questions | form six Pure Advanced Mathematics

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#	Question
1	If $underline{a}=3underline{i}-5underline{j}-2underline{k}$ and $underline{b}=7underline{i}+underline{j}-2underline{k}$ are non zero vectors, Find the projection of $underline{a}$ onto $underline{b}$ . Use vectors to prove the sine rule. If $heta$ is the angle between two unit vectors $underline{a}$ and $underline{b}$ show that $frac{1}{2}\|underline{a}+underline{b}\|= cosleft ( frac{ heta}{2} ight )$ If $small G(t)=e^{t}underline{i}+ cost underline{j}+ tunderline{k}$ find $small frac{d}{dt}[(sin t)G(t)]$ Integrate the vector $small e^{t}underline{i}+2tunderline{j}+ ln tunderline{k}$ with respect to t. The two vectors $small underline{a}$ and $small underline{b}$ have the same magnitude and an angle between them is $small 60degree$ . If their scalar product is $small frac{1}{2}$ , find their magnitude. Mathematical Calculation
2	Find the unit vector perpendicular to both vectors $small underline{a}=3underline{i}+2underline{j}+2underline{k}$ and $small underline{b}=underline{i}+2underline{j}-2underline{k}$ . Mathematical Calculation