If `vec(A) = 4hat(i) - 2hat(j) + 6hat(k)` and `B = hat(i) - 2hat(j) - 3hat(k)` the angle which the `vec(A) vec(B)` makes with x-axis is :

Options

A. `cos^(-1)((1)/(10))`
B. `cos^(-1)((2)/(sqrt(10)))`
C. `cos^(-1)((1)/(sqrt(10)))`
D. `cos^(-1)((1)/(5))`

Answer

To find the angle that the vector \(\vec{A} - \vec{B}\) makes with the x-axis, we will follow these steps:

### Step 1: Define the vectors
Given:
\[ \vec{A} = 4\hat{i} - 2\hat{j} + 6\hat{k} \]
\[ \vec{B} = \hat{i} - 2\hat{j} - 3\hat{k} \]

### Step 2: Calculate the vector \(\vec{A} - \vec{B}\)
To find \(\vec{A} - \vec{B}\), we subtract the components of \(\vec{B}\) from \(\vec{A}\):
\[ \vec{A} - \vec{B} = (4\hat{i} - 2\hat{j} + 6\hat{k}) - (\hat{i} - 2\hat{j} - 3\hat{k}) \]
Breaking it down:
- For the \(\hat{i}\) component: \(4 - 1 = 3\)
- For the \(\hat{j}\) component: \(-2 - (-2) = 0\)
- For the \(\hat{k}\) component: \(6 - (-3) = 9\)

Thus, we have:
\[ \vec{A} - \vec{B} = 3\hat{i} + 0\hat{j} + 9\hat{k} \]

### Step 3: Identify the components of the resulting vector
The resulting vector can be expressed as:
\[ \vec{C} = 3\hat{i} + 0\hat{j} + 9\hat{k} \]
Where:
- \(C_x = 3\)
- \(C_y = 0\)
- \(C_z = 9\)

### Step 4: Find the angle with the x-axis
The angle \(\theta\) that the vector makes with the x-axis can be found using the formula:
\[ \tan(\theta) = \frac{C_z}{C_x} \]
Substituting the values:
\[ \tan(\theta) = \frac{9}{3} = 3 \]

### Step 5: Calculate \(\theta\)
To find \(\theta\):
\[ \theta = \tan^{-1}(3) \]

### Step 6: Find the cosine of the angle
To express the angle in terms of cosine, we can use the relationship:
\[ \cos(\theta) = \frac{C_x}{\sqrt{C_x^2 + C_z^2}} \]
Calculating:
\[ \cos(\theta) = \frac{3}{\sqrt{3^2 + 9^2}} = \frac{3}{\sqrt{9 + 81}} = \frac{3}{\sqrt{90}} = \frac{3}{3\sqrt{10}} = \frac{1}{\sqrt{10}} \]

### Step 7: Final angle with respect to x-axis
Thus, the angle \(\theta\) with respect to the x-axis is:
\[ \theta = \cos^{-1}\left(\frac{1}{\sqrt{10}}\right) \]

### Summary
The angle which the vector \(\vec{A} - \vec{B}\) makes with the x-axis is:
\[ \theta = \cos^{-1}\left(\frac{1}{\sqrt{10}}\right) \]