KEAM 2011 Math Paper

Let a unit vector in XOY -plane is,

∧

a

=‌

a

|a|

a=a1i+a2j,|a|=√a12+a22
Given vectors let b=i+j
and ‌‌c=i−j
From question;

∧

a

⋅b=|

∧

a

|⋅|b|cos‌30∘
=1⋅√2⋅‌

√3

2

⇒‌

√6

2

⇒

a⋅b

|a|

=‌

√6

2

⇒a1+a2=‌

√6

2

|a| .....(i)
and

∧

a

⋅c=|

∧

a

||c|⋅cos‌60∘
=1⋅√2⋅‌

1

2

‌

a⋅c

|a|

=‌

√2

2

⇒a1−a2=‌

√2

2

|a|.....(ii)
From Eqs. (i) and (ii),
2a1=(√6+√2)‌

|a|

2

a1=‌

1

4

(√6+√2)|a|
⇒

a1

|a|

=‌

1

4

(√6+√2)
and a2=‌

1

4

(√6−√2)|a|
⇒

a2

|a|

=‌

1

4

(√6−√2)
So, the unit vector is

∧

a

=‌

1

|a|

(a1i+a2j)

∧

a

=‌

1

4

{(√6+√2)i+(√6−√2)j}