This is a practice question I have for a test and have absolutely no idea how to solve...
Given Information:
1) Vector p is a unit vector
2) Vector q is any non-zero vector
3) Vector r = (p dot q)(p)
4) Vector s = q - r
Prove that p and s are perpendicular using the properties of vectors.
On
If you define $<x,y>$ as the dot product, to show that $x$ and $y$ are perpendicular, you need to show that
$<s,p>=0$
Then, you can plug in the formula that you have for $s$ and use the linearity of dot product, which says
$<\alpha x+\beta y,z>=\alpha<x,z>+\beta <y,z>$
where $\alpha, \beta$ are scalars and $x,y,z$ are vectors.
Hope you are willing to do the rest.
we have $$p\cdot s=p\cdot(q-r)=p\cdot (q-(p\cdot q)*p)=p\cdot q-(p\cdot q)*p^2=p\cdot q-p\cdot q=0$$ since $$p^2=1$$ it is a unit vector