Question

If vectors \( \vec{a} = 3\hat{i} + 2\hat{j} + \lambda\hat{k} \) and \( \vec{b} = 2\hat{i} - 4\hat{j} + 5\hat{k} \) represent the two strips of the Red Cross sign placed outside a doctor's clinic, then the value of \( \lambda \) is :

• A. \( 1 \)
• B. \( \frac{5}{2} \)
• C. \( \frac{2}{5} \)
• D. \( 0 \)

Hint:

Real-world application word cues like "Red Cross sign" or "orthogonal axes" indicate that the underlying angle is exactly \( 90^\circ \), meaning you should set the dot product equal to zero.

Answer 1

The Correct Option is B

Concept: The geometry of a standard Red Cross sign consists of two perpendicular straight strips that intersect at right angles (\( 90^\circ \)). For any two vectors that are perpendicular, their dot product equals zero: \[ \vec{a} \cdot \vec{b} = 0 \]

Step 1: Express the condition for perpendicular vectors. Since the strips represent the Red Cross sign, the vectors \( \vec{a} \) and \( \vec{b} \) must be orthogonal to one another: \[ \vec{a} \cdot \vec{b} = 0 \]

Step 2: Expand the dot product using components. The vectors are given as: \[ \vec{a} = 3\hat{i} + 2\hat{j} + \lambda\hat{k} \] \[ \vec{b} = 2\hat{i} - 4\hat{j} + 5\hat{k} \] Compute the dot product by multiplying corresponding component parts: \[ (3)(2) + (2)(-4) + (\lambda)(5) = 0 \] \[ 6 - 8 + 5\lambda = 0 \]

Step 3: Solve for the unknown variable \( \lambda \). Combine the constants: \[ -2 + 5\lambda = 0 \] Isolate the term containing \( \lambda \): \[ 5\lambda = 2 \implies \lambda = \frac{2}{5} \] Let's double check the calculation steps: \( 3 \times 2 = 6 \); \( 2 \times (-4) = -8 \); \( 6 - 8 = -2 \); \( 5\lambda = 2 \implies \lambda = 2/5 \). This matches option (C) perfectly.