Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that $\vec{a} \cdot \vec{b}=0$. For some $x, y \in \mathbb{R}$, let $\vec{c}=x \vec{a}+y \vec{b}+(\vec{a} \times \vec{b})$. If $|\vec{c}|=2$ and the vector $\vec{c}$ is inclined at the same angle $\alpha$ to both $\vec{a}$ and $\vec{b}$, then the value of $8 \cos ^{2} \alpha$ is ____.

JEE Advanced · 2018 · 20 May 2018 · Paper 1 · Vector Algebra

Let

a

and

b

be two unit vectors such that

a \cdot b = 0

. For some

x, y \in R

, let

c = x a + y b + (a \times b)

. If

∣ c ∣ = 2

and the vector

c

is inclined at the same angle

α

to both

a

and

b

, then the value of

8 cos^{2} α

is ____.

Answer: 3

Since

a

and

b

are orthogonal unit vectors,

∣ a \times b ∣ = 1

c \cdot a = x ∣ a ∣^{2} = x

, and

∣ c ∣ cos α = x

, so

2 cos α = x

.

Similarly,

c \cdot b = y

, so

2 cos α = y

.

Thus

x = y = 2 cos α

∣ c ∣^{2} = x^{2} + y^{2} + ∣ a \times b ∣^{2} = 4 cos^{2} α + 4 cos^{2} α + 1 = 4

.

So

8 cos^{2} α = 3

JEE Advanced · 2018 · 20 May 2018 · Paper 1 · Numerical answer