Are Vectors v-w and v+w Perpendicular When Magnitudes of v and w Are Equal?

[andrassy]

Homework Statement

Theres two questions I need help with on m homework:

I need to prove algebraically that the linear system r + 2s = -b1 and 3r+5s = b2 has a solution for all numbers b1, b2

also: for vectors v and w prove that v-w and v+w are perpendicular if and only if the magnitude of v equals the magnitude of w.

The Attempt at a Solution

The first question i multiplied the first equation by -3 and added the two equations together to get 11s = b2 - 3b1 but i have no idea where to go from there.

The second I proved that two vectors are perpendicular if their dot product is zero. I did the dot product of v-w and v+w an dgot [v1^2 - w1^2, . . . vn^2 - wn^2].here agian in stuck. any help please?

 

[Defennder]

When you say prove algebraically, are you allowed to use matrices here? It's a lot easier if you could do so. Start by representing the linear system as an augmented matrix:

$$\left(\begin{array}{*{20}c}1&2&-b_{1}\\3&5&b_{2}\end{array}\right)$$

If you want to show that there are exactly 1 solutions for both r,s , you need to show that you can reduce the augmented matrix (the sub-matrix on the left) above to the identity matrix.

For the 2nd part, your approach is correct, but you should get this:

$$(v+w)\cdot(v-w) = v\cdot v + v\cdot w - w\cdot v - w\cdot w$$

You know where to go from here, right?

 

[Rainbow Child]

The first question i multiplied the first equation by -3 and added the two equations together to get 11s = b2 - 3b1 but i have no idea where to go from there.

You are almost done. Solve for s, and sustitude the result in one of original the equations in order to find r.

for vectors v and w prove that v-w and v+w are perpendicular if and only if the magnitude of v equals the magnitude of w.

Write the dot product $(\vec{v}-\vec{w})\cdot (\vec{v}+\vec{w})=0$ and expand it.