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Question
If `|bar"a".bar"b"| = |bar"a" xx bar"b"|` and `bar"a".bar"b" < 0`, then find the angle between `bar"a" "and" bar"b"`.
Solution
Let θ be the angle between `bar"a"` and `bar"b"`
Then `|bar"a".bar"b"| = |bar"a" xx bar"b"|` gives
`|"ab cos" theta| = |"ab sin" theta|`
∴ - ab cos θ = ab sin θ
∴ - 1 = tan θ
∴ tan θ = - tan `pi/4` = tan `(pi - pi/4)`
∴ tan θ = tan `(3pi)/4`
∴ `theta = (3pi)/4`
Hence, the angle between `bar"a"` and `bar"b"` is `(3pi)/4`.
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