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Question
If A, B, C, D are the points with position vectors `hat"i" + hat"j" - hat"k", 2hat"i" - hat"j" + 3hat"k", 2hat"i" - 3hat"k", 3hat"i" - 2hat"j" + hat"k"`, respectively, find the projection of `vec"AB"` along `vec"CD"`.
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Solution
Here, Position vector of A = `hat"i" + hat"j" - hat"k"`
Position vector of B = `2hat"i" - hat"j" + 3hat"k"`
Position vector of C = `2hat"i" - 3hat"k"`
Position vector of D = `3hat"i" - 2hat"j" + hat"k"`
`vec"AB"` = P.V of B – P.V of A
= `(2hat"i" - hat"j" + 3hat"k") - (hat"i" + hat"j" - hat"k")`
= `hat"i" - 2hat"j" + 4hat"k"`
`vec"CD"` = P.V. of D – P.V. of C
= `(3hat"i" - 2hat"j" + hat"k") - (2hat"i" - 3hat"k")`
= `hat"i" - 2hat"j" + 4hat"k"`
Projection of `vec"AB"` on `vec"CD" = (vec"AB" * vec"Cd")/|vec"CD"|`
= `((hat"i" - 2hat"j" + 4hat"k") * (hat"i" - 2hat"j" + 4hat"k"))/sqrt((1)^2 + (-2)^2 + (4)^2)`
= `(1 + 4 + 16)/sqrt(1 + 4 + 16)`
= `21/sqrt(21)`
= `sqrt(21)`
Hence, the required projection = `sqrt(21)`.
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