JEE Main 2025 Apr 4 Shift 1, Mathematics Q9: Scalar (or Dot) Product of Two Vectors

JEE Main2025Apr 4, Shift 1Mathematics

Let $A$ and $B$ be two distinct points on the line $L : \frac{x - 6}{3} = \frac{y - 7}{2} = \frac{z - 7}{- 2}$ . Both $A$ and $B$ are at a distance $217$ from the foot of perpendicular drawn from the point $(1, 2, 3)$ on the line $L$ . If $O$ is the origin, then $O A \cdot O B$ is equal to:

A
49
B
47
C
21
D
62

Solution

$P Q \cdot b = 0$

$\Rightarrow 3 (3 λ + 5) + 2 (2 λ + 5) - 2 (- 2 λ + 4) = 0$

$\Rightarrow 17 λ = - 17$

$\Rightarrow λ = - 1$

$Q = (3, 5, 9)$

Let $A = (3 μ + 6, 2 μ + 7, - 2 μ + 7)$ .

Since $O A = 68$ ,

$(3 μ + 6)^{2} + (2 μ + 7)^{2} + (- 2 μ + 7)^{2} = 68$

$\Rightarrow μ^{2} + 2 μ - 3 = 0$

$\Rightarrow μ = - 3, 1$

$∴ A = (- 3, 1, 13)$ and $B = (9, 9, 5)$

$O A \cdot O B = (- 3) (9) + (1) (9) + (13) (5) = 47$

$47$

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JEE Main 2025 Apr 4 Shift 1, Mathematics Q9: Scalar (or Dot) Product of Two Vectors

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