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Question
If `vec"a" = 2hat"i" + 3hat"j" - 4hat"k", vec"b" = 3hat"i" - 4hat"j" - 5hat"k"`, and `vec"c" = -3hat"i" + 2hat"j" + 3hat"k"`, find the magnitude and direction cosines of `3vec"a"- 2vec"b"+ 5vec"c"`
Solution
`3vec"a"- 2vec"b"+ 5vec"c" = 3(2hat"i" + hat"j" - 4hat"k") -2(3hat"i" - 4hat"j" - 5hat"k") + 5(-3hat"i" + 2hat"j" + 3hat"k")`
= `6hat"i" + 9hat"j" - 12hat"k" - 6hat"i" + 8hat"j" + 10hat"k" - 15hat"i" + 10hat"j" + 15hat"k"`
`3vec"a"- 2vec"b"+ 5vec"c" = -15hat"i" + 27hat"j" + 13hatk"`
`|3vec"a"- 2vec"b"+ 5vec"c"| = |-15hat"i" + 27hat"j" + 13hatk"|`
= `sqrt((-1)^2 + (27)^2 + 13^2`
=`sqrt(225 + 729 + 169)`
`|3vec"a"- 2vec"b"+ 5vec"c"| = sqrt(1123)`
Direction cosines of the vector `3vec"a"- 2vec"b"+ 5vec"c"` are
`[(-15)/|-15hat"i" + 27hat"j" + 1hat"k"|, 27/|-15hat"i" + 27hat"j" + 13hat"k"|, 13/|-15hat"i" + 27hat"j" + 13hat"k"|`
`[(-15)/sqrt(113), 27/sqrt(1123), 13/sqrt(123)]`
∴ The magnitude and direction cosines of the vector `3vec"a"- 2vec"b"+ 5vec"c"` are
`sqrt(1123), [(-15)/sqrt(113), 27/sqrt(1123), 13/sqrt(123)]`
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