Let a^{\to} = i^{\^}+j^{\^}+k^{\^} b^{\to} = i^{\^}-j^{\^}+k^{\^} and c^{\to} = i^{\^}-j^{\^}-k^{\^} be three vectors. A vector v^{\to} in the plane of a^{\to} and b^{\to} whose projection on c^{\to}/ {|c^{\to}|} is 1/\sqrt3 , is :

Correct Answer is : 3i^−j^+3k^

NIMCET 2015 Question Paper with solutions

Question : 10 of 120

Marks: +1, -0

Let

a↖{→}

i↖{\^}+j↖{\^}+k↖{\^}

b↖{→}

i↖{\^}-j↖{\^}+k↖{\^}

and

c↖{→}

i↖{\^}-j↖{\^}-k↖{\^}

be three vectors. A vector

v↖{→}

in the plane of

a↖{→}

and

b↖{→}

whose projection on

c↖{→}/ {|c↖{→}|}

1/√3

, is :

Solution:

We have

a=i↖{\^}+j↖{\^}+k↖{\^}

b=i↖{\^}-j↖{\^}+k↖{\^}

and

c=i↖{\^}-j↖{\^}-k↖{\^}

A vector v in plane of a and b

∴v=λa+µb

⇒=λ(i↖{⋀}+j↖{⋀}+k↖{⋀})+µ(i↖{⋀}-j↖{⋀}+k↖{⋀})

=(λ+µ)i↖{⋀}+(λ-µ)j↖{⋀}+(λ+µ)k↖{⋀}

Projection of v and

c/{|c|}

1/√3

∴{(λ+µ)i↖{⋀}+(λ-µ)j↖{⋀}+(λ+µ)k↖{⋀}⋅(i↖{⋀}-j↖{⋀}-k↖{⋀}){/}√{3}}/{|{c}/{|c|}|}={1}/{√{3}}

{(λ+µ)-(λ-µ)-(λ+µ)}/{√{3}}={1}/{√{3}}

⇒λ-µ=-1

∴ Option (a) is correct

Go to Question:

Prev Question

Next Question