I have three vectors: vec(u) , vec(v) and vec(w) and this equality: vec(u) * vec(v) =vec(u) * vec(w). Then the question is if vec(v) =vec(w).

Nathanial Levine 2022-10-01 Answered
I have three vectors: u , v and w and this equality: u · v = u · w . Then the question is if v = w . At first i tried to write the equality representing components:
u · v = u 1 · v 1 + + u n · v n
u · w = u 1 · w 1 + + u n · w n
Then:
u 1 · v 1 + + u n · v n = u 1 · w 1 + + u n · w n .
But I got stuck right there; I thought if c i , i [ 1 , n ] could be canceled from both terms, but don't know if that is possible somehow.
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Answers (2)

Frida King
Answered 2022-10-02 Author has 8 answers
Let u = v = 0 and w 0 then u v = u w = 0
More generally u v = u w u ( v w ) = 0 u perpendicular to ( v w )
And the above example is true as 0 is perpendicular to every vector.
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st3he1d0t
Answered 2022-10-03 Author has 3 answers
You can rearrange and use associativity of the dot product:
u v = u w u ( v w ) = 0
Which implies that:
The vector (of nonzero norm) formed by the difference v w is orthogonal to u (also of nonzero norm)
Or
u is a null vector
AND/OR
v w is a null vector.
The final condition implies v = w , but it is clearly not the only possibility.
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