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Question
If `vec"a", vec"b", vec"c"` are three vectors such that `vec"a" + vec"b" + vec"a" = vec0` and `|vec"a"|` = 2, `|vec"b"|` = 3, `|vec"c"|` = 5, then value of `vec"a"*vec"b" + vec"b"*vec"c" + vec"c"*vec"a"` is ______.
Options
0
1
– 19
38
Solution
If `vec"a", vec"b", vec"c"` are three vectors such that `vec"a" + vec"b" + vec"a" = vec0` and `|vec"a"|` = 2, `|vec"b"|` = 3, `|vec"c"|` = 5, then value of `vec"a"*vec"b" + vec"b"*vec"c" + vec"c"*vec"a"` is – 19.
Explanation:
Given that `|vec"a"|` = 2, `|vec"b"|` = 3, `|vec"c"|` = 5
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"b" + |vec"c"|^2` = 0
⇒ `|vec"a"|^2 + |vec"b"|^2 + |vec"c"|^2 + 2vec"a"*vec"b" + 2vec"b"*vec"c" + 2vec"c"*vec"a"` = 0
⇒ `(2)^2 + (3)^2 + (5)^2 + 2(vec"a"*vec"b" + vec"b"*vec"c" + vec"c"*vec"a")` = 0
⇒ `4 + 9 + 25 + 2 (vec"a"*vec"b" + vec"b"*vec"c" + vec"c"*vec"a")` = 0
⇒ `38 + 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")` = – 38
∴ `vec"a"*vec"b" + vec"b"*vec"c" + vec"c"*vec"a"` = – 19
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