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Question
If `vec"a" = hat"i" + hat"j" + 2hat"k"` and `vec"b" = 2hat"i" + hat"j" - 2hat"k"`, find the unit vector in the direction of `6vec"b"`
Solution
Given that `vec"a" = hat"i" + hat"j" + 2hat"k"` and `vec"b" = 2hat"i" + hat"j" - 2hat"k"`
`6vec"b" = 6(2hat"i" + hat"j" - 2hat"k")`
= `12hat"i" + 6hat"j" - 12hat"k"`
∴ Unit vector in the direction of `6vec"b" = (6vec"b")/|6vec"b"|`
= `(12hat"i" + 6hat"j" - 12hat"k")/sqrt((12)^2 + (6)^2 + (-12)^2)`
= `(12hat"i" + 6hat"j" - 12hat"k")/sqrt(144 + 36 + 144)`
= `(12hat"i" + 6hat"j" - 12hat"k")/sqrt(324)`
= `(12hat"i" + 6hat"j" - 12hat"k")/18`
= `6/18 (2hat"i" + hat"j" - 2hat"k")`
= `1/3(2hat"i" + hat"j" - 2hat"k")`
Hence, the required unit vector is `1/3(2hat"i" + hat"j" - 2hat"k")`.
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