# Prove That 2 Vectors are Perpendicular Using The Properties of Vectors

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.

• which theorems can you use? – Dr. Sonnhard Graubner Nov 30 '16 at 15:53
• This is a grade 12 calculus and vectors class, so we only know the basics ie. Cross product, dot product, vector addition and vector subtraction. We havent learned any theorums (yet) – Mark Dodds Nov 30 '16 at 15:54
• i would use the dot product – Dr. Sonnhard Graubner Nov 30 '16 at 15:55
• for 3) its the dot product of p and q multiplied by vector p – Mark Dodds Nov 30 '16 at 15:56
• p dot q gives a number – Dr. Sonnhard Graubner Nov 30 '16 at 15:56

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

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.