Question

If ä and b are two vectors 'such that
à .b = 0 and ã ×b =0 then which one
of the following is correct?
(a) a is parallel to b.
(b) a is perpendicular to b
(c) Either a or b is a null vector.
(d) None of the above​

Answer 1

Given : - a . b = 0 ; a x b = 0

Formulae : -  a . b = |a| x |b| cos∅  ;  a x b = |a| x |b| sin∅

Solution : -

a . b = 0 ⇒ |a| x |b| cos∅ = 0

⇒ |a| x |b| = 0 or cos∅ = 0

Also, a x b = 0 ⇒ |a| x |b| sin∅ = 0

⇒ |a| x |b| = 0   or   sin∅

∵ sin∅  and cos∅  both cannot be equal to zero simultaneously

⇒ |a| x |b| = 0

∴ (c) either a or b is a null vector.

Correct answer is : - option (c) Either a or b is a null vector.