Question

If two vectors are A = 2hati + hatj - hatk and B = hatj - 4hatk. By calculation, prove AxxB isperpendicular to both A and B.

Answer 1

Given :

A = 2i + j - k

B = j - 4k

To prove :

A×B is perpendicular to both A & B

Proof :

•If two vectors A & B are in the form of A = ai + bj + ck & B = di + ej + fk

then cross product of two vectors i.e. A × B is given by

•A × B = i j k

a b c

d e f

So,

•A × B = i j k

2 1 -1

0 1 -4

•A × B = i [ (1)(-4)-(1)(-1) ] -j [ (2)(-4) -(0)(-1) ] + k [ (2)(1) - (0)(-1) ]

•A × B = -3i + 8j +2k

•Now if two vectors are perpendicular to each other then, their dot product will be zero

Because A.B = |A||B|cosQ

•where Q is angle between vector A and Vector B

•so , if two vectors are perpendicular to each other then Cos Q will be zero

• So, (A×B).A = (-3)(2) + (1)(8) + (-1)(2)

= -6+8-2 = 0

•This means A×B is perpendicular to A

•Similarly ,

(A × B).B = (-3)(0) + (1)(8) + (-4)(2)

= 8-8 = 0

•This means A×B is perpendicular to Vector B

•Hence proved , that A × B is perpendicular to both vector A & vector B