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