# Thread: [SOLVED] Quick Communication Vector Question

1. ## [SOLVED] Quick Communication Vector Question

a) If u and v are non-zero vectors, but Proj(u onto v) = 0, what conclusion can be drawn?
b) If Proj(u onto v) = 0, does it follow that Proj(v onto u) = 0? Explain.

Small, clear explanation would be fine thanks

2. $v \ne 0\;\& \,proj_v u = \frac{{u \cdot v}}{{v \cdot v}}v\, \Rightarrow \,u \cdot v = 0$

3. ## Vector and scalar product

Hello narutoblaze
Originally Posted by narutoblaze
a) If u and v are non-zero vectors, but Proj(u onto v) = 0, what conclusion can be drawn?
b) If Proj(u onto v) = 0, does it follow that Proj(v onto u) = 0? Explain.

Small, clear explanation would be fine thanks
(a) Use the scalar (dot) product formula to give the projection of $\vec{u}$ onto $\vec{v}$, which is:

proj( $\vec{u}$ onto $\vec{v}$) $=\frac{\vec{u}\cdot \vec{v}}{|\vec{v}|} = 0, |\vec{v}| \ne 0$

$\Rightarrow \vec{u}\cdot \vec{v} = 0$

$\Rightarrow |\vec{u}||\vec{v}|\cos\theta = 0$, where $\theta$ is the angle between the vectors $\vec{u}$ and $\vec{v}$

$\Rightarrow \theta = \tfrac{\pi}{2}$, since $\vec{u}$ and $\vec{v}$ are non-zero.

So the vectors are perpendicular.

(b) Yes: proj( $\vec{v}$ onto $\vec{u}$)= $\frac{\vec{u}\cdot \vec{v}}{|\vec{u}|} = 0, |\vec{u}| \ne 0$

$\Rightarrow \frac{\vec{u}\cdot \vec{v}}{|\vec{v}|} = 0$

$\Rightarrow$ proj( $\vec{u}$ onto $\vec{v}$)= 0