Dot Product
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New answer posted
4 weeks agoContributor-Level 10
|3a+4b|² = 9|a|² + 16|b|² + 24a·b
But a·b = 0, |a|=|b|=k
|3a+4b| = 5k
|4a-3b|
10k = 20? k = 2 = |a| = |b|
New answer posted
a month agoContributor-Level 10
Given r * a = r * b, which means r * a - r * b = 0 ⇒ r * (a - b) = 0.
This implies that vector r is parallel to vector (a - b).
So, r = λ (a - b) for some scalar λ.
a - b = (2i - 3j + 4k) - (7i + j - 6k) = -5i - 4j + 10k.
So, r = λ (-5i - 4j + 10k).
We are also given r ⋅ (i + 2j + k) = -3.
λ (-5i - 4j + 10k) ⋅ (i + 2j + k) = -3
λ (-51 - 42 + 10*1) = -3
λ (-5 - 8 + 10) = -3
λ (-3) = -3 ⇒ λ = 1.
So, r = 1 * (-5i - 4j + 10k) = -5i - 4j + 10k.
We need to find r ⋅ (2i - 3j + k).
(-5i - 4j + 10k) ⋅ (2i - 3j + k) = (-5) (2) + (-4) (-3) + (10) (1)
= -10 + 12 + 10 = 12.
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