JEE Main 2025 Apr 8 Shift 2, Mathematics Q9: Applications of Vectors

JEE Main2025Apr 8, Shift 2Mathematics

Let $a = \hat{i} + 2 \hat{j} + \hat{k}$ and $b = 2 \hat{i} + \hat{j} - \hat{k}$ . Let $\overset{c}{^}$ be a unit vector in the plane of the vectors $a$ and $b$ and be perpendicular to $a$ . Then such a vector $\overset{c}{^}$ is:

A
$\frac{1}{5} (\hat{j} - 2 \hat{k})$
B
$\frac{1}{3} (- \hat{i} + \hat{j} - \hat{k})$
C
$\frac{1}{3} (\hat{i} - \hat{j} + \hat{k})$
D
$\frac{1}{2} (- \hat{i} + \hat{k})$

Solution

Any vector in the plane of $a$ and $b$ has the form $p = a + λ b$ for some scalar $λ$ .

Perpendicularity to $a$ requires $p \cdot a = 0$ , i.e. $a \cdot a + λ (b \cdot a) = 0$ .

Compute $a \cdot a = 1 + 4 + 1 = 6$ and $a \cdot b = 2 + 2 - 1 = 3$ , so $6 + 3 λ = 0$ , giving $λ = - 2$ .

Then $p = (1, 2, 1) - 2 (2, 1, - 1) = (- 3, 0, 3)$ , with magnitude $32$ .

The unit vector is $\overset{c}{^} = \frac{( - 3 , 0 , 3 )}{3 2} = \frac{- i ^ + k ^}{2}$ , matching option (4).

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JEE Main 2025 Apr 8 Shift 2, Mathematics Q9: Applications of Vectors

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