JEE Advanced 2024 Paper 2, Mathematics Section 3 Q4: Vector (or Cross) Product of Two Vectors

JEE Advanced2024Paper 2Mathematics Section 3

Let $p = 2 \hat{i} + \hat{j} + 3 \hat{k}$ and $q = \hat{i} - \hat{j} + \hat{k}$ . If for some real numbers $α$ , $β$ and $γ$ , we have

$15 \hat{i} + 10 \hat{j} + 6 \hat{k} = α (2 p + q) + β (p - 2 q) + γ (p \times q),$

then the value of $γ$ is ______.

Your answer

Solution

Compute the three building blocks.

$2 p + q = (4 + 1) \hat{i} + (2 - 1) \hat{j} + (6 + 1) \hat{k} = 5 \hat{i} + \hat{j} + 7 \hat{k} .$

$p - 2 q = (2 - 2) \hat{i} + (1 + 2) \hat{j} + (3 - 2) \hat{k} = 0 \hat{i} + 3 \hat{j} + \hat{k} .$

$p \times q = \hat{i} \hat{j} \hat{k} 213 1 - 11 = \hat{i} (1 + 3) - \hat{j} (2 - 3) + \hat{k} (- 2 - 1) = 4 \hat{i} + \hat{j} - 3 \hat{k} .$

Matching components on each side of $15 \hat{i} + 10 \hat{j} + 6 \hat{k} = α (5, 1, 7) + β (0, 3, 1) + γ (4, 1, - 3)$ :

$5 α + 4 γ = 15, α + 3 β + γ = 10, 7 α + β - 3 γ = 6.$

Solving this linear system yields $α = \frac{7}{5}$ , $β = \frac{11}{5}$ , $γ = 2$ .

$∴ γ = 2.$

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