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Question
If `vec"a" = hat"i" + hat"j" + 2hat"k"` and `hat"b" = 2hat"i" + hat"j" - 2hat"k"`, find the unit vector in the direction of `2vec"a" - vec"b"`
Solution
Given that `vec"a" = hat"i" + hat"j" + 2hat"k"` and `hat"b" = 2hat"i" + hat"j" - 2hat"k"`
`2vec"a" - vec"b" = 2(hat"i" + hat"j" + 2hat"k") - (2hat"i" + hat"j" - 2hat"k")`
= `2hat"i" + 2hat"j" + 4hat"k" - 2hat"i" - hat"j" + 2hat"k"`
= `hat"j" + 6hat"k"`
∴ Unit vector in the direction of `2vec"a" - vec"b"`
= `(2vec"a" - vec"b")/|2vec"a" - vec"b"|`
= `(hat"j" + 6hat"k")/sqrt((1)^2 + (6)^2)`
= `(hat"j" + 6hat"k")/sqrt(1 + 36)`
= `(hat"j" + 6hat"k")/sqrt(37)`
= `1/sqrt(37) [hat"j" + 6hat"k"]`
Hence, the required unit vector is `1/sqrt(37) [hat"j" + 6hat"k"]`.
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