Question

If the vector \(\vec{a}=x\hat{i}-\hat{j}+\hat{k}\) and \(\vec{b}=\hat{i}+y\hat{j}+z\hat{k}\) are collinear, then a possible unit vector parallel to the vector \(\hat{i}+y\hat{j}+z\hat{k}\) is...

• A. \(\frac{1}{\sqrt{3}}(\hat{i}-\hat{j}+\hat{k})\)
• B. \(\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})\)
• C. \(\frac{1}{\sqrt{3}}(\hat{i}-\hat{j}-\hat{k})\)
• D. \(\frac{1}{\sqrt{3}}(-\hat{i}-\hat{j}-\hat{k})\)

Hint:

Collinear vectors have proportional components; use this to relate y and z.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
If \(\vec{a}\) and \(\vec{b}\) are collinear, then \(\vec{b}=t\vec{a}\) for some scalar \(t\).

Step 2: Compare components
\(1=tx\), \(y=-t\), \(z=t\). Hence \(y=-z\).

Step 3: Form of the vector
The vector \(\hat{i}+y\hat{j}+z\hat{k}\) is \(\hat{i}-t\hat{j}+t\hat{k}\), so its \(\hat{j}\) and \(\hat{k}\) components are opposite.

Step 4: Compare the options
Only \(\dfrac{1}{\sqrt{3}}(\hat{i}-\hat{j}+\hat{k})\) has opposite \(\hat{j}\) and \(\hat{k}\) components; it corresponds to \(t=1\), that is \(x=1\).

Final Answer:
A possible unit vector is one over root 3 times (i - j + k). \[ \boxed{\text{(A) }\dfrac{1}{\sqrt{3}}(\hat{i}-\hat{j}+\hat{k})} \]