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Prove that two vectors must have equal magnitude if their sum is perpendicular to their difference?
Wiki User
∙ 12y ago
Suppose the condition stated in this problem holds for the two vectors a and b. If the
sum a+b is perpendicular to the difference a-b then the dot product of these two vectors
is zero:
(a + b) · (a - b) = 0
Use the distributive property of the dot product to expand the left side of this equation. We
get:
a · a - a · b + b · a - b · b
But the dot product of a vector with itself gives the magnitude squared:
a · a = a2
x + a2
y + a2
z = a2
(likewise b · b = b2) and the dot product is commutative: a · b = b · a. Using these facts,
we then have
a2 - a · b + a · b + b2 = 0 ,
which gives:
a2 - b2 = 0 =) a2 = b2
Since the magnitude of a vector must be a positive number, this implies a = b and so vectors
a and b have the same magnitude.
Wiki User
∙ 12y ago
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