Mathematics - Vector Question with Solution | TestHub

Given$\overrightarrow{a}=\alpha \hat{i}+3\hat{j}-\hat{k},\overrightarrow{b}=3\hat{i}-\beta \hat{j}+4\hat{k}$and$\overrightarrow{c}=\hat{i}+2\hat{j}-2\hat{k}$where$\alpha ,\beta \in R$be three vectors.

Projection of $\overrightarrow{a}$ on $\overrightarrow{c}$ is $\frac{\overrightarrow{a}·\overrightarrow{c}}{\left|\overrightarrow{c}\right|}=\frac{10}{3}$ (given)

$\Rightarrow \frac{\alpha +6+2}{\sqrt{1+4+4}}=\frac{10}{3}\Rightarrow \alpha =2$

and $\overrightarrow{b}\times \overrightarrow{c}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 3& -\beta & 4\\ 1& 2& -2\end{array}\right|=-6\hat{i}+\hat{j}+\hat{k}$

$\Rightarrow 2\beta -8=-6\Rightarrow \beta =1$

Hence $\alpha +\beta =3$