Question

The vector `vec"a" + vec"b"` bisects the angle between the non-collinear vectors `vec"a"` and `vec"b"` if ______.

Answer 1

The vector `vec"a" + vec"b"` bisects the angle between the non-collinear vectors `vec"a"` and `vec"b"` if `vec"a" = vec"b"`.

Explanation:

If vector `vec"a" + vec"b"` bisects the angle between non-collinear vectors `vec"a"` and `vec"b"` then the angle between `vec"a" + vec"b"` and `vec"a"` is equal to the angle between `vec"a" + vec"b"` and `vec"b"`.

So, `cos  theta = (vec"a" * (vec"a" + vec"b"))/(|vec"a"||vec"a" + vec"b"|)`

= `(vec"a" * (vec"a" + vec"b"))/(|vec"a"| sqrt("a"^2 + "b"^2))`  ......(i)

Also, `cos theta = (vec"b"*(vec"a" + vec"b"))/(|vec"b"|*|vec"a" + vec"b"|)`  .....`[because theta  "is same"]`

= `(vec"b" * (vec"a" + vec"b"))/(|vec"b"| sqrt("a"^2 + "b"^2))`  ......(ii)

From equation (i) and equation (ii) we get,

`(vec"a" * (vec"a" + vec"b"))/(|vec"a"| sqrt("a"^2 + "b"^2)) = (vec"b" * (vec"a" + vec"b"))/(|vec"b"| sqrt("a"^2 + "b"^2))`

⇒ `vec"a"/|vec"a"| = vec"b"/|vec"b"|`

⇒ `hat"a" = hat"b"`

⇒ `vec"a" = vec"b"`

Hence, the required filler is `vec"a" = vec"b"`.