Maharashtra State BoardHSC Arts 12th Board Exam
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If |a¯⋅b¯|=|a¯×b¯| and a¯⋅b¯<0, then find the angle between a¯ and b¯ - Mathematics and Statistics

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Sum

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")`

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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`.

Concept: Vector Product of Vectors (Cross)
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