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.

  • $\begingroup$ which theorems can you use? $\endgroup$ – Dr. Sonnhard Graubner Nov 30 '16 at 15:53
  • $\begingroup$ 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) $\endgroup$ – Mark Dodds Nov 30 '16 at 15:54
  • $\begingroup$ i would use the dot product $\endgroup$ – Dr. Sonnhard Graubner Nov 30 '16 at 15:55
  • $\begingroup$ for 3) its the dot product of p and q multiplied by vector p $\endgroup$ – Mark Dodds Nov 30 '16 at 15:56
  • $\begingroup$ p dot q gives a number $\endgroup$ – 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


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.


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