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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 `vec"a", vec"b", vec"c"`
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Solution
`vec"a", vec"b", vec"c" = (2hat"i" + 3hat"j" - 4hat"k") + (3hat"i" - 4hat"j" - 5hat"k") + (-3hat"i" + 2hat"j" + 3hat"k")`
`vec"a", vec"b", vec"c" = 2hat"i" + hat"j" - 6hat"k"`
`|vec"a", vec"b", vec"c"| = |2hat"i" + hat"j" - 6hat"k"|`
= `sqrt(2^2 + 1^2 + (-6)^2`
= `sqrt(4 + 1 + 36)`
= `sqrt(41)`
Direction cosnes of `2hat"i" + hat"j" - 6hat"k"` are
`[2/|2hat"i" + hat"j" - 6hat"k"|, 1/|2hat"i" + hat"j" - 6hat"k"|, (-6)/|2hat"i" + hat"j"- 6hat"k"|]`
`[2/sqrt(41), 1/sqrt(41), (6)/sqrt(41)]`
∴ he magnitde and direction cosines of the vector.
`vec"a" + vec"b" + vec"c"` are `sqrt(41), [2/sqrt(41), 1/sqrt(41), (6)/sqrt(41)]`
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