If a vector A is given as `A = 4 hati +3hatj+12hatk` , then the angle subtended with the x - axis is

Options

A. `sin^(-1)[4/13]`
B. `sin^(-1)[3/13]`
C. `cos^(-1)[3/13]`
D. `cos^(-1)[4/13]`

Answer

To find the angle subtended by the vector \( \mathbf{A} = 4 \hat{i} + 3 \hat{j} + 12 \hat{k} \) with the x-axis, we can follow these steps:

### Step 1: Identify the components of the vector
The vector \( \mathbf{A} \) has the following components:
- \( A_x = 4 \) (the component along the x-axis)
- \( A_y = 3 \) (the component along the y-axis)
- \( A_z = 12 \) (the component along the z-axis)

### Step 2: Calculate the magnitude of the vector
The magnitude of the vector \( \mathbf{A} \) can be calculated using the formula:
\[ |\mathbf{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2} \]
Substituting the values:
\[ |\mathbf{A}| = \sqrt{4^2 + 3^2 + 12^2} = \sqrt{16 + 9 + 144} = \sqrt{169} = 13 \]

### Step 3: Use the cosine formula to find the angle
The angle \( \theta \) subtended with the x-axis can be found using the cosine formula:
\[ \cos \theta = \frac{A_x}{|\mathbf{A}|} \]
Substituting the values:
\[ \cos \theta = \frac{4}{13} \]

### Step 4: Calculate the angle \( \theta \)
To find \( \theta \), we take the inverse cosine:
\[ \theta = \cos^{-1}\left(\frac{4}{13}\right) \]

### Final Answer
The angle subtended by the vector \( \mathbf{A} \) with the x-axis is:
\[ \theta = \cos^{-1}\left(\frac{4}{13}\right) \]

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