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Question
If `vec"a", vec"b", vec"c"` are unit vectors such that `vec"a" + vec"b" + vec"c"` = 0, then the value of `vec"a" * vec"b" + vec"b" * vec"c" + vec"c" * vec"a"` is ______.
Options
1
3
` -3/2`
None of these
Solution
If `vec"a", vec"b", vec"c"` are unit vectors such that `vec"a" + vec"b" + vec"c"` = 0, then the value of `vec"a" * vec"b" + vec"b" * vec"c" + vec"c" * vec"a"` is ` -3/2`.
Explanation:
Given that: `|vec"a"| = |vec"b"| = |vec"c"|` = 1
And `vec"a" + vec"b" + vec"c" = vec0`
∴ `(vec"a" + vec"b" + vec"c") * (vec"a" + vec"b" + vec"c") = vec0 * vec0` = 0
`|vec"a"|^2 + vec"a" * vec"b" + vec"a" * vec"c" + vec"b" * vec"a" + |vec"b"|^2 + vec"b" * vec"c" + vec"c" * vec"a" + vec"c" + vec"b" + |vec"c"|^2` = 0
⇒ `|vec"a"|^2 + |vec"b"|^2 * |vec"c"|^2 +2 vec"a" * vec"b" + 2vec"b" * vec"c" + 2vec"c" * vec"a"` = 0
⇒ `1 + 1 + 1 + 2(vec"a" * vec"b" + vec"b" * vec"c" + vec"c" * vec"a")` = 0
⇒ `2(vec"a" * vec"b" + vec"b" * vec"c" + vec"c" * vec"a")` = – 3
⇒ `vec"a" * vec"b" + vec"b" * vec"c" + vec"c" * vec"a" = (-3)/2`
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