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Question
Find the angle between the vectors `2hat"i" + hat"j" - hat"k"` and `hat"i" + 2hat"j" + hat"k"` using vector product
Solution
Let the given vector be `2hat"i" + hat"j" - hat"k"` and `hat"i" + 2hat"j" + hat"k"`
`vec"a" xx vec"b" = |(hat"i", hat"j", hat"k"),(2, 1, -1),(1, 2, 1)|`
= `hat"i"(1 + 2) - hat"j"(2 + 1) + hat"k"(4 - 1)`
`vec"a" xx vec"b" = 3hat"i" - 3hat"j" + 3hat"k"`
`|vec"a" xx vec"b"| = |3hat"i" - 3hat"j" + 3hat"k"|`
= `sqrt(3^2 + (- 3)^2 + 3^2`
= `sqrt(3 xx 3^2)`
= `3sqrt(3)`
`|vec"a"| = |2hat"i" + hat"j" - hat"k"|`
= `sqrt(2^2 + 1^2 + (- 1)^2`
= `sqrt(4 + 1 + 1)`
= `sqrt(6)`
`|vec"b"| = |hat"i" + 2hat"j" - hat"k"|`
= `sqrt(1^2 + 2^2 + 1^2)`
= `sqrt(1 + 4 + 1)`
= `sqrt(6)`
Let θ be the angle between `vec"a"` and `vec"b"`
sin θ = `|vec"a" xx vec"b"|/(|vec"a"| |vec"b"|)`
= `(3sqrt(3))/(sqrt(6) * sqrt(6))`
= `(3sqrt(3))/6`
sin θ = `sqrt(3)/2`
∴ θ = `pi/3`
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