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If vec(a)=hat(i)+2hat(j)-3hat(k) and ve...

If `vec(a)=hat(i)+2hat(j)-3hat(k)` and `vec(b)=3hat(i)-hat(j)+2hat(k)`, then show that `(vec(a)+vec(b))` is perpendicular to `(vec(a)-vec(b))`.

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Given `vec(a)= hat(i) + 2hat(j)- 3hat(k) and vec(b) = 3hat(i) - hat(j)+ 2hat(k)`
`therefore vec(a) + vec(b)= (hat(i) + 2hat(j)- 3hat(k))+ (3hat(i)- hat(j) + 2hat(k))= 4hat(i) + hat(j)- hat(k)`
And `vec(a)- vec(b)= (hat(i) + 2hat(j)- 3hat(k))- (3hat(i) - hat(j) + 2hat(k))= -2hat(i) + hat(j)- 5hat(k)`
`therefore (vec(a)+vec(b)). (vec(a)- vec(b))= 4(-2) + 1 xx 3 + (-1) (-5)= -8 +3+5= 0`
Therefore, the vectors `vec(a) + vec(b) and vec(a)- vec(b)` are perpendicular to each other.
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