Question

Find a vector of magnitude $5$ units which is parallel to the vector $2\hat{i}-\hat{j}$.

Answer 1

$\theta ={\cos }^{-1}\left(\frac{v_1.v_2}{|v_1|.|v_2|}\right)$

$v_1=x+y\hat{j}$

$v_2=2i-\hat{j}$

$|v_1|=5$

$|v_2|=\sqrt{4+1}=\sqrt{5}$

$v_1+v_2$ are parallel

then $\theta =180$.

$\cos 180=\frac{(x\hat{i}+y\hat{i})(2\hat{i}-\hat{j})}{5.\sqrt{5}}$

$-1=\frac{2x-y}{2\sqrt{5}}=2x-y=5\sqrt{5}$ ---- $\left(1\right)$

$v_1$ and $v_2$ are parallel then.

$x_1.y_2=x_2{y}_1\Rightarrow x.(-1)=y\left(2\right)\Rightarrow 2y=-x$

$-4y-y=5\sqrt{5}x=-2y$

$-5y=5\sqrt{5}\Rightarrow y=-\sqrt{5}x=2\sqrt{5}$

$v_1=2\sqrt{5}i-\sqrt{5}\hat{j}$