JEE Main · 2021 · 20 Jul 2021 · Shift 1 · Vector Algebra

Let

a, b, c

be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle

θ

, with the vector

a + b + c

. Then

36 cos^{2} 2 θ

is equal to ____

Answer: 4

Solution

Let the common magnitude be

k

. Mutual perpendicularity gives

a + b + c^{2} = ∣ a ∣^{2} + ∣ b ∣^{2} + ∣ c ∣^{2} = 3 k^{2}

and

a \cdot (a + b + c) = ∣ a ∣^{2} = k^{2}

, the same for

b

and

c

, which is why the three inclinations are equal. Hence

cos θ = \frac{k ^{2}}{k \cdot k 3} = \frac{1}{3}

cos 2 θ = 2 cos^{2} θ - 1 = \frac{2}{3} - 1 = - \frac{1}{3}

and

36 cos^{2} 2 θ = 36 (\frac{1}{9}) = 4

JEE Main · 2021 · 20 Jul 2021 · Shift 1 · Integer answer