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:

Perpendicular vectors always have a zero dot product. This is the most common way to find an unknown parameter in vector questions involving geometry.

Answer 1

The Correct Option is C

Concept:
• The Red Cross sign consists of two rectangular strips that are perpendicular to each other.
• If two non-zero vectors \( \vec{a} \) and \( \vec{b} \) are perpendicular, their dot product must be zero: \( \vec{a} \cdot \vec{b} = 0 \).
• Dot product of \( a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \) and \( b_1\hat{i} + b_2\hat{j} + b_3\hat{k} \) is \( a_1b_1 + a_2b_2 + a_3b_3 \).

Step 1: Identify the geometric relationship between the vectors
The question mentions the "Red Cross sign".
In a Red Cross sign, the two strips intersect at right angles (\( 90^\circ \)).
Therefore, the vectors \( \vec{a} \) and \( \vec{b} \) representing these strips are perpendicular to each other.

Step 2: Apply the condition for perpendicularity
For \( \vec{a} \perp \vec{b} \), we have:
\[ \vec{a} \cdot \vec{b} = 0 \]
Substitute the components of the given vectors:
\[ (3\hat{i} + 2\hat{j} + \lambda\hat{k}) \cdot (2\hat{i} - 4\hat{j} + 5\hat{k}) = 0 \]

Step 3: Solve for \( \lambda \)
Multiply the corresponding components:
\[ (3 \times 2) + (2 \times -4) + (\lambda \times 5) = 0 \]
\[ 6 - 8 + 5\lambda = 0 \]
\[ -2 + 5\lambda = 0 \]
\[ 5\lambda = 2 \]
\[ \lambda = \frac{2}{5} \]