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Question
Find the unit vector in the direction of the sum of the vectors `vec"a" = 2hat"i" - hat"j" + 2hat"k"` and `vec"b" = -hat"i" + hat"j" + 3hat"k"`.
Solution
Let `vec"c"` denote the sum of `vec"a"` and `vec"b"`.
We have `vec"c" = (2hat"i" - hat"j" + 2hat"k") + (-hat"i" + hat"j" + 3hat"k")`
= `hat"i" + 5hat"k"`
Now `|vec"c"| = sqrt(1^2 + 5^2)`
= `sqrt(26)`
Thus, the required unit vector is `hat"c" = vec"c"/|vec"c"| = 1/sqrt(26)(hat"i" + 5hat"k")`
= `1/sqrt(26) hat"i" + 5/sqrt(26) hat"k"`.
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