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) 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
Hello narutoblaze(a) Use the scalar (dot) product formula to give the projection of $\displaystyle \vec{u}$ onto $\displaystyle \vec{v}$, which is:
proj($\displaystyle \vec{u}$ onto $\displaystyle \vec{v}$)$\displaystyle =\frac{\vec{u}\cdot \vec{v}}{|\vec{v}|} = 0, |\vec{v}| \ne 0$
$\displaystyle \Rightarrow \vec{u}\cdot \vec{v} = 0$
$\displaystyle \Rightarrow |\vec{u}||\vec{v}|\cos\theta = 0$, where $\displaystyle \theta$ is the angle between the vectors $\displaystyle \vec{u}$ and $\displaystyle \vec{v}$
$\displaystyle \Rightarrow \theta = \tfrac{\pi}{2}$, since $\displaystyle \vec{u}$ and $\displaystyle \vec{v}$ are non-zero.
So the vectors are perpendicular.
(b) Yes: proj($\displaystyle \vec{v}$ onto $\displaystyle \vec{u}$)=$\displaystyle \frac{\vec{u}\cdot \vec{v}}{|\vec{u}|} = 0, |\vec{u}| \ne 0$
$\displaystyle \Rightarrow \frac{\vec{u}\cdot \vec{v}}{|\vec{v}|} = 0$
$\displaystyle \Rightarrow$ proj($\displaystyle \vec{u}$ onto $\displaystyle \vec{v}$)= 0
Grandad