Question

What is the relation between the two vectors \(\vec{A}=2\hat{i}+5\hat{j}-6\hat{k}\) and \(\vec{B}=\hat{i}+\frac{5}{2}\hat{j}-3\hat{k}\) ?

• A. \(\vec{A}\) is perpendicular to \(\vec{B}\)
• B. \(\vec{A}\) is parallel to \(\vec{B}\)
• C. \(\vec{A}+\vec{B}=0\)
• D. \(\vec{A}-\vec{B}=0\)

Hint:

Compare the components of the two vectors and check whether one is a constant multiple of the other.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Two vectors are parallel when one is a scalar multiple of the other, so their components are in the same ratio.

Step 2: Calculation
\(\vec{A}=2\hat{i}+5\hat{j}-6\hat{k}\) and \(\vec{B}=\hat{i}+\frac{5}{2}\hat{j}-3\hat{k}\). Ratios of components: \(\frac{2}{1}=2\), \(\frac{5}{5/2}=2\), \(\frac{-6}{-3}=2\). So \(\vec{A}=2\vec{B}\).

Step 3: Compare the options
Since \(\vec{A}=2\vec{B}\) with a positive factor, the vectors point the same way, so they are parallel. \(\vec{A}\cdot\vec{B}=2+\frac{25}{2}+18\neq 0\), so they are not perpendicular, and neither the sum nor the difference is zero.

Final Answer:
The two vectors are parallel, because \(\vec{A}=2\vec{B}\). \[ \boxed{\text{(B) }\vec{A}\ \text{is parallel to}\ \vec{B}} \]