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Question
Find the sine of the angle between the vectors `vec"a" = 3hat"i" + hat"j" + 2hat"k"` and `vec"b" = 2hat"i" - 2hat"j" + 4hat"k"`.
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Solution
Given that `vec"a" = 3hat"i" + hat"j" + 2hat"k"` and `vec"b" = 2hat"i" - 2hat"j" + 4hat"k"`.
We know that `|vec"a" xx vec"b"| = |vec"a"||vec"b"| sin theta`
∴ `vec"a" xx vec"b" = |(hat"i", hat"j", hat"k"),(3, 1, 2),(2, -2, 4)|`
= `hat"i"(4 + 4) - hat"j"(12 - 4) + hat"k"(-6 - 2)`
= `8hat"i" - 8hat"j" - 8hat"k"`
`|vec"a" xx vec"b"| = sqrt((8)^2 + (-8)^2 + (-8)^2)`
= `sqrt(64 + 64 + 64)`
= `sqrt(192)`
= `sqrt(64 xx 3)`
= `8sqrt(3)`
`|vec"a"| = sqrt((3)^2 + (1)^2 + (2)^2)`
= `sqrt(9 + 1 + 4)`
= `sqrt(14)`
`|vec"b"| = sqrt((2)^2 + (-2)^2 + (4)^2)`
= `sqrt(4 + 4 + 16)`
= `sqrt(24)`
= `2sqrt(6)`
∴ `sin theta = |vec"a" xx vec"b"|/(|vec"a"||vec"b"|)`
= `(8sqrt(3))/(sqrt(14) * 2sqrt(6))`
⇒ `(4sqrt(3))/sqrt(84) = (4sqrt(3))/(2sqrt(21))`
= `2/sqrt(7)`
Hence, `sin theta = 2/sqrt(7)`.
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