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Question
Find the unit vector parallel to `3vec"a" - 2vec"b" + 4vec"C"` if `vec"a" = 3hat"i" - hat"j" - 4hat"k", vec"b" = -2hat"i" + 4hat"j" - 3hat"k"`, and `vec"c" = hat"i" + 2hat"j" - hat"k"`
Solution
Given that `vec"a" = 3hat"i" - hat"j" - 4hat"k"`
`vec"b" = -2hat"i" + 4hat"j" - 3hat"k"`
`vec"c" = hat"i" + 2hat"j" - hat"k"`
`3vec"a" - 2vec"b" + 4vec"c" = 3(3hat"i" - hat"j" - 4hat"k") -2(-2hat"i" + 4hat"j" - 3hat"k") + 4(hat"i" + 2hat"j" - hat"k")`
= `9hat"i" - 3hat"j" - 12hat"k" + 4hat"i" - 8hat"j" + 6hat"k" + 4hat"i" + 8hat"j" - 4hat"k"`
`3vec"a" - 2vec"b" + 4vec"c" = 17hat"i" - 3hat"j" - 10hat"k"`
The unit vector parallel to `3vec"a" - 2vec"b" + 4vec"c"` is
= `(3vec"a" - 2vec"b" + 4vec"c")/|3vec"a" - 2vec"b" + 4vec"c"|`
= `(17hat"i" - 3hat"j" - 10hat"k")/|17hat"i" - 3hat"j" - 10hat"k"|`
= `(17hat"i" - 3hat"j" - 10hat"k")/sqrt(17^2 + (-3)^2 + 10)^2`
= `(17hat"i" - 3hat"j" - 10hat"k")/sqrt(289 + 9 + 100)`
= `(17hat"i" - 3hat"j" - 10hat"k")/sqrt(398)`
Thus, the unut vector parallel to `3vec"a" - 2vec"b" + 4vec"c"` is
= `(17hat"i" - 3hat"j" - 10hat"k")/sqrt(398)`
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