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Let hat"a", hat"b", hat"c" be unit vectors such that hat"a".hat"b" = hat"a".hat"c" = 0 and 6  the angle between hat"b" and hat"c" is pi/6. Prove that hat"a" = +- 2(hat"b" xx hat"c"). - Mathematics and Statistics

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Sum

Let hat"a", hat"b", hat"c" be unit vectors such that hat"a".hat"b" = hat"a".hat"c" = 0 and 6  the angle between hat"b" and hat"c" is pi/6. Prove that hat"a" = +- 2(hat"b" xx hat"c").

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Solution

`hat"a".hat"b" = hat"a".hat"c" = 0`

∴ `hat"a"` is perpendicular to `hat"b" and hat"c"` both

∴ `hat"a"` is parallel to `hat"b" xx hat"c"`

∴ `hat"a" = "m"(hat"b" xx hat"c")`, m is a scalar.

∴ `|hat"a"| = |"m"| |hat"b" xx hat"c"|`

∴ `|hat"a"| = |"m"| |hat"b"||hat"c"|`

∴ `|hat"a"| = |"m"| |hat"b" xx hat"c"| sin pi/6`

∴ `1 = |"m"| xx 1 xx 1 xx 1/2 = |"m"|/2     .....[because |hat"a"| = |hat"b"| = |hat"c"| = 1]`

∴ 2 = |m|

∴ m = ± 2

∴ `hat"a" = ± 2(hat"b" xx hat"c")`.

Concept: Representation of Vector
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