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 ______.

विकल्प

• 1

• 3

• ` -3/2`

• None of these

Answer 1

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`