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Question
If `vec"a" = 2hat"i" - hat"j" + hat"k", vec"b" = hat"i" + hat"j" - 2hat"k"` and `vec"c" = hat"i" + 3hat"j" - hat"k"`, find `lambda` such that `vec"a"` is perpendicular to `lambdavec"b" + vec"c"`.
Solution
We have `lambda vec"b" + vec"c" = lambda (hat"i" + hat"j" - 2hat"k") + (hat"i" + 3hat"j" - hat"k")`
= `(lambda + 1)hat"i" + (lambda + 3)hat"j" - (2lambda + 1)hat"k"`
Since `vec"a"` ⊥ `(lambdavec"b" + vec"c"), vec"a"*(lambda vec"b" + vec"c")` = 0
⇒ `(2hat"i" - hat"j" + hat"k") * [(lambda + 1)hat"i" + (lambda + 3)hat"j" - (2lambda + 1)hat"k"]` = 0
⇒ `2(lambda + 1) - (lambda + 3) - (2lambda + 1)` = 0
⇒ `lambda` = – 2
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