Vector Algebra

Let

→

a

=

^

i

+

^

j

+2

^

k

,

→

b

=

^

i

+2

^

j

+

^

k

and

→

c

=

^

i

+

^

j

+

^

k

Any vector in the plane of

^

i

+

^

j

+2

^

k

and

^

i

+2

^

j

+

^

k

is given by

→

r

= λ

→

a

+µ

→

b

= λ(

^

i

+

^

j

+2

^

k

)+µ(

^

i

+2

^

j

+

^

k

)
= (λ+µ)

^

i

+(λ+2µ)

^

j

+(2λ+µ)

^

k

Also ,

→

r

.

→

c

= 0
⇒ (λ + µ) . 1 + (λ + 2µ) . 1 + (2λ + µ) . 1 = 0
⇒ 4λ + 4µ = 0
⇒ λ + µ = 0
⇒ [

→ r

→ a

→ b

] = 0
So, vectors

^

j

−

^

k

and −

^

j

+

^

k

satisfy this.