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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
We know that,
`bar("a")*bar("b") = |bar("a")| |bar("b")|` cos θ
∴ `|bar("a")*bar("b")| = ||bar("a")| |bar("b")| cosθ|`
∴ `|bar("a")*bar("b")| = |bar("a")| |bar("b")|` cos θ ......(i) `[bar("a")*bar("b") < 0]`
Also, `|bar("a") xx bar("b")| = |bar("a")| |bar("b")|` sin θ .......(ii)
`|bar("a")*bar("b")| = |bar("a") xx bar("b")|` .......[Given]
∴ `-|bar("a")| |bar("b")| cos theta = |bar("a")| |bar("b")| sin theta` .......[From (i) and (ii)]
∴ −1 = tan θ
∴ tan θ = −1
∴ θ = `tan^-1(-1) = (3pi^"c")/4`
∴ The angle between `bar("a")` and `bar("b")` is `(3pi^"c")/4`.
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