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.

Answer 1

`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.