If \( \vec{a}=\hat{\imath}-7\hat{\jmath}+4\hat{k} \) and \( \vec{b}=2\hat{\imath}-14\hat{\jmath}-p\hat{k} \) are parallel , then the value of \( p \), is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → −8
Two vectors are parallel when their components are proportional. The k-component of \( \vec{b} \) is −p:
\( \displaystyle \frac{2}{1}=\frac{-14}{-7}=\frac{-p}{4} \)
Both first ratios equal 2, so
\( \displaystyle -\frac{p}{4}=2 \)
\( \displaystyle p=-8 \)
Check: with p = −8, \( \vec{b}=2\hat{\imath}-14\hat{\jmath}+8\hat{k}=2\vec{a} \).
Why the other options are incorrect
Option (1) −1/8 — Incorrect. With \( p=-\frac{1}{8} \), the k-component of \( \vec{b} \) is \( \frac{1}{8} \), and \( \frac{1/8}{4}\ne 2 \).
Option (2) 8 — Incorrect. This misses the minus sign in \( -p\hat{k} \): with p = 8, the k-component of \( \vec{b} \) is −8, and \( \frac{-8}{4}=-2\ne 2 \).
Option (4) 1/8 — Incorrect. With \( p=\frac{1}{8} \), the k-component of \( \vec{b} \) is \( -\frac{1}{8} \), and \( \frac{-1/8}{4}\ne 2 \).