The vector a^{\to} = \alphai^{\^}+2j^{\^}+\betak^{\^} lies in the plane of the vectors b^{\to} = i^{\^}+j^{\^} and c^{\to} = j^{\^}+k^{\^} and bisects the angle between b^{\to} and c^{\to} . Then, which one of the following gives possible values of α and β?

Correct Answer is : α = 1 , β = 1

NIMCET 2018 Question Paper with solutions

Question : 23 of 120

Marks: +1, -0

The vector

a↖{→}

αi↖{\^}+2j↖{\^}+βk↖{\^}

lies in the plane of the vectors

b↖{→}

i↖{\^}+j↖{\^}

and

c↖{→}

j↖{\^}+k↖{\^}

and bisects the angle between

b↖{→}

and

c↖{→}

. Then, which one of the following gives possible values of α and β?

Solution:

Now, unit vector along angle bisector of

b↖{→}

and

c↖{→}

d↖{→}

1/2(b↖{→}/{|b↖{→}|}+c↖{→}/{|c↖{→}|})

1/2({i↖{\^}+j↖{\^}}/√{1^2+1^2}+{j↖{\^}+k↖{\^}}/√{1^2+1^2})

1/{2√2} (i↖{\^}+2j↖{\^}+k↖{\^})

Since,

a↖{→}

is along

d↖{→}

,
∴

a↖{→}

= λ

d↖{→}

⇒

αi↖{\^}+2j↖{\^}+βk↖{\^}

λ/{2√2} (i↖{\^}+2j↖{\^}+k↖{\^})

On comparing the coefficient of i , j and k, we get
λ =

2√2

∴ α = 1 and β = 1

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