Question

The angle made by a vector \(\vec{B}=3\hat{i}+2\hat{j}+4\hat{k}\) with Y axis is

• A. \(\cos ^{-1}\left(\frac{4}{\sqrt{11}}\right)\)
• B. \(\cos ^{-1}\left(\frac{3}{\sqrt{17}}\right)\)
• C. \(\cos ^{-1}\left(\frac{2}{\sqrt{29}}\right)\)
• D. \(\cos ^{-1}\left(\frac{5}{\sqrt{23}}\right)\)

Hint:

The angle with an axis comes from the direction cosine, the component along that axis divided by the magnitude of the vector.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
The angle a vector makes with an axis follows from its direction cosine.
For the Y axis, \(\cos\beta=\dfrac{B_y}{|\vec{B}|}\).

Step 2: Calculation
The magnitude is \(|\vec{B}|=\sqrt{3^2+2^2+4^2}=\sqrt{29}\).
The Y component is \(B_y=2\).

Step 3: Result
\(\cos\beta=\dfrac{2}{\sqrt{29}}\), so \(\beta=\cos^{-1}\left(\dfrac{2}{\sqrt{29}}\right)\).

Step 4: Compare the options
This expression matches option (C).

Final Answer:
The vector makes an angle of cos inverse of 2 by root 29 with the Y axis. \[ \boxed{\text{(C) }\cos^{-1}\left(\dfrac{2}{\sqrt{29}}\right)} \]