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Question
Let `vec("a") = hat"i" + 2hat"j" - 3hat"k"` and `vec("b") = 3hat"i" -"j" +2hat("k")` be two vectors. Show that the vectors `(vec("a")+vec("b"))` and `(vec("a")-vec("b"))`are perpendicular to each other.
Solution
`vec("a") = hat"i" + 2hat"j" - 3hat"k"`
`vec("b") = 3hat"i" - hat"j" + 2hat"k"`
`vec("a") + vec("b") = (hat"i" + 2hat"j" - 3hat"k") + (3hat"i"-hat"j" + 2hat"k")`
`= (4hat"i" + hat"j" - hat("k"))`
`vec("a") - vec("b") = (hat"i" + 2hat"j" -3hat"k") - (3hat"i" - hat"j"+ 2hat"k")`
` = (-2hat"i" + 3hat"j" - 5hat"k")`
`(vec("a")+ vec("b")).(vec("a")-vec("b")) = (4hat"i" + hat"j" - hat"k") . (-2hat"i" + 3hat"j" -5hat"k")`
= - 8 + 3 +5
= 0
Since the dot product of `(vec("a")+vec("b"))` and`(vec("a")- vec("b"))` is 0 so, they are perpendicular to each other.
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