This question is about vectors: "Find the projection of $v=2i+3j$ in the direction of $w=4i-2j$."
I'm not exactly sure how I should approach this problem. I've been told I have to use dot product, but that didn't help as much. Can someone show me how I'm supposed to solve this problem?
On
The projection of $\underline{a}$ onto $\underline{b}$ is $$\frac{\underline{a}\cdot\underline{b}}{|\underline{b}|}$$
On
For any vectors $\vec u, \vec v\neq 0$, the horizontal magnitude of $\vec v$ on the axis of $\vec u$ is $|\vec v|\cos\theta$, where $\theta$ is the angle between the 2 vectors.
For the direction of the projection vector, it is just the unit vector of $\vec u$, that is $\displaystyle \frac{\vec u}{|\vec u|}$.
Therefore, the projection vector is $\displaystyle \frac{\vec u}{|\vec u|}\cdot|\vec v|\cos\theta=\frac{\vec u\cdot\vec v}{|\vec u|^2}\vec u$.
And the length is $\displaystyle\frac{\vec u\cdot\vec v}{|\vec u|^2}\cdot|\vec u|=\frac{\vec u\cdot\vec v}{|\vec u|}$.
Since, $$V.W=\Vert V\,\Vert\Vert W\Vert\,Cos\theta$$ and $$V\,Cos\theta=Projection\,of\,V\,in\,the\,direction\,of\,W$$ Therfore, $$V.W=\Vert W\Vert V_w$$ $$V_w=\frac{V.W}{\Vert W\Vert}$$ So, yo have to find dot product of $V$ and $W$, then divide it by the magnitude of $W$.