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Question
Find the unit vectors perpendicular to each of the vectors `vec"a" + vec"b"` and `vec"a" - vec"b"`, where `vec"a" = hat"i" + hat"j" + hat"k"` and `vec"b" = hat"i" + 2hat"j" + 3hat"k"`
Solution
Th given vectors are `vec"a" = hat"i" + hat"j" + hat"k"` and `vec"b" = hat"i" + 2hat"j" + 3hat"k"`
`(vec"a" + vec"b") xx (vec"a" - vec"b") = vec"a" xx vec"a" - vec"a" xx vec"b" + vec"b" xx vec"a" - vec"b" xx vec"b"`
= `0 - vec"a" xx vec"b"- vec"a" xx vec"b" - 0`
= `- 2 vec"a" xx vec"b"`
`vec"a" xx vec"b" = |(hat"i", hat"j", hat"k"),(1, 1, 1),(1, 2, 3)|`
= `hat"i"(3 - 2) - hat"j"(3 - 1) + hat"k"(2 - 1)`
`vec"a" xx vec"b" = hat"i" - 2hat"j" + hat"k"`
`|vec"a" xx vec"b"| = |hat"i" - 2hat"j" + hat"k"|`
= `sqrt(1^2 + (-2)^2 + 1^2)`
= `sqrt(1 + 4 + 1)`
= `sqrt(6)`
The unit vector perpendicular to `vec"a" + vec"b"` and
`vec"a" - vec"b"` is = `+- ((vec"a" + vec"b") xx (vec"a" - vec"b"))/|(vec"a" + vec"b") xx (vec"a" - vec"b")|`
= `+- (-2(vec"a" xx vec"b"))/(|-2(vec"a" xx vec"b")|)`
= `+- (-2(hat"i" - 2hat"j" + hat"k"))/(2 xx sqrt(6))`
= `+- ((-hat"i" + 2hat"j" - hat"k"))/sqrt(6)`
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