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JEE Main · 2023 · 13 Apr 2023 · Shift 2 · Vector Algebra

Let

∣ a ∣ = 2

,

∣ b ∣ = 3

and the angle between the vectors

a

and

b

be

\frac{π}{4}

. Then

∣ (a + 2 b) \times (2 a - 3 b) ∣^{2}

is equal to

(A)482
(B)441
(C)841
(D)882✓ Correct

Answer: (D)

Solution

Expanding the cross product:

(a + 2 b) \times (2 a - 3 b) = 2 a \times a - 3 a \times b + 4 b \times a - 6 b \times b

Since

a \times a = b \times b = 0

and

b \times a = - a \times b

:

= - 3 a \times b - 4 a \times b = - 7 a \times b

The magnitude is:

∣ - 7 a \times b ∣ = 7∣ a ∣∣ b ∣ sin \frac{π}{4} = 7 \cdot 2 \cdot 3 \cdot \frac{1}{2} = \frac{42}{2} = 212

Squaring:

∣ - 7 a \times b ∣^{2} = (212)^{2} = 441 \cdot 2 = 882

JEE Main · 2023 · 13 Apr 2023 · Shift 2 · Single correct option

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