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JEE Main · 2023 · 31 Jan 2023 · Shift 2 · Vector Algebra

Let

a, b, c

be three vectors such that

∣ a ∣ = 31, 4∣ b ∣ = ∣ c ∣ = 2

and

2 (a \times b) = 3 (c \times a)

. If the angle between

b

and

c

is

\frac{2 π}{3}

, then

(\frac{a \times c}{a \cdot b})^{2}

is equal to

Answer: 3

Solution

Since

2 (a \times b) = 3 (c \times a) = - 3 (a \times c)

,

a \times (2 b + 3 c) = 0

, so

a = λ (2 b + 3 c) .

With

∣ b ∣ = \frac{1}{2}, ∣ c ∣ = 2, b \cdot c = \frac{1}{2} \cdot 2 cos \frac{2 π}{3} = - \frac{1}{2}

,

∣2 b + 3 c ∣^{2} = 4 \cdot \frac{1}{4} + 9 \cdot 4 + 12 (- \frac{1}{2}) = 31

, and

∣ a ∣^{2} = 31

forces

λ^{2} = 1.

Then

a \times c = 2 (b \times c)

with

∣ b \times c ∣ = \frac{1}{2} \cdot 2 sin \frac{2 π}{3} = \frac{3}{2}

, so

∣ a \times c ∣ = 3

, while

a \cdot b = λ (2∣ b ∣^{2} + 3 c \cdot b) = λ (\frac{1}{2} - \frac{3}{2}) = - λ

, giving

∣ a \cdot b ∣ = 1.

Hence

(\frac{a \times c}{a \cdot b})^{2} = \frac{3}{1} = 3.

JEE Main · 2023 · 31 Jan 2023 · Shift 2 · Integer answer

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