Question 1206236
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I'll use a period to indicate dot product.
A . B = A dot product B
where A and B are vectors.


Here are the dot product rules to memorize<ul><li>A.B = B.A</li><li>A.(B+C) = A.B+A.C</li><li>0.A = 0</li><li>|A| = sqrt(A.A)</li><li>(k*A).B = k*(A.B) = A.(k*B) where k is a scalar and * means the usual multiplication</li></ul>Let
S = A+B
represent the sum of the vectors


Recall that vectors are perpendicular if and only if their dot product is zero.
We'll start with the idea A+B is perpendicular to A-B.
That must mean (A + B) . (A - B) = 0 
The goal is to end up with |A| = |B| to show they have the same magnitude.


(A + B) . (A - B) = 0
S . (A - B) = 0
S . A - S . B = 0 
S . A = S . B
(A+B) . A = (A+B) . B
A . (A+B) = B . (A+B)
A.A + A.B = B.A + B.B
A.A + A.B = B.B + A.B
A.A = B.B
sqrt(A.A) = sqrt(B.B)
|A| = |B|


We have shown that (A + B) . (A - B) = 0 leads to |A| = |B|
Therefore, if A+B is perpendicular to A-B, then vectors A and B must have the same magnitude.
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