Question:
If \(\vec{a} = 2\hat{i} + 3\hat{j} + 4\hat{k}\) and \(\vec{b} = 3\hat{i} + 2\hat{j} – \lambda\hat{k}\) are perpendicular, then what is the value of \(\lambda\)?
यदि \(\vec{a} = 2\hat{i} + 3\hat{j} + 4\hat{k}\) और \(\vec{b} = 3\hat{i} + 2\hat{j} – \lambda\hat{k}\) परस्पर लंबवत हैं, तो \(\lambda\) का मान क्या है?
Options:
- 2
- 3
- 4
- 5
- 2
- 3
- 4
- 5
Correct Answer: b)
Explanation:
Two vectors \(\vec{a}\) and \(\vec{b}\) are perpendicular if and only if their scalar dot product is zero:
\[\vec{a} \cdot \vec{b} = 0\]
\[(2)(3) + (3)(2) + (4)(-\lambda) = 0\]
\[6 + 6 – 4\lambda = 0 \implies 12 – 4\lambda = 0 \implies \lambda = 3\]
Hence, the correct option is (b).
दो सदिश \(\vec{a}\) और \(\vec{b}\) परस्पर लंबवत होते हैं यदि उनका अदिश गुणनफल शून्य हो:
\[\vec{a} \cdot \vec{b} = 0\]
\[(2)(3) + (3)(2) + (4)(-\lambda) = 0\]
\[6 + 6 – 4\lambda = 0 \implies 12 = 4\lambda \implies \lambda = 3\]
अतः सही विकल्प (b) है।