Given `vec(A) = hat(i) - 2hat(j) - 3hat(k) , vec(B) = 4hat(i) - 2hat(j) + 6hat(k)` .Calculate the angle made by `(vec(A) +vec(B))` with the x - axis ?

Explanation

To solve the problem of finding the angle made by the vector \(\vec{C} = \vec{A} + \vec{B}\) with the x-axis, we will follow these steps:

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

### Step 2: Calculate \(\vec{C} = \vec{A} + \vec{B}\)
We will add the corresponding components of \(\vec{A}\) and \(\vec{B}\):
\[ \vec{C} = \vec{A} + \vec{B} = (\hat{i} + 4\hat{i}) + (-2\hat{j} - 2\hat{j}) + (-3\hat{k} + 6\hat{k}) \]
\[ \vec{C} = (1 + 4)\hat{i} + (-2 - 2)\hat{j} + (-3 + 6)\hat{k} \]
\[ \vec{C} = 5\hat{i} - 4\hat{j} + 3\hat{k} \]

### Step 3: Find the components of \(\vec{C}\)
From \(\vec{C}\), we have:
- \(C_x = 5\)
- \(C_y = -4\)
- \(C_z = 3\)

### Step 4: Calculate the magnitude of \(\vec{C}\)
The magnitude of \(\vec{C}\) is given by:
\[ |\vec{C}| = \sqrt{C_x^2 + C_y^2 + C_z^2} \]
Substituting the values:
\[ |\vec{C}| = \sqrt{5^2 + (-4)^2 + 3^2} = \sqrt{25 + 16 + 9} = \sqrt{50} \]

### Step 5: Calculate the cosine of the angle \(\alpha\) with the x-axis
The cosine of the angle \(\alpha\) made by \(\vec{C}\) with the x-axis is given by:
\[ \cos \alpha = \frac{C_x}{|\vec{C}|} \]
Substituting the values:
\[ \cos \alpha = \frac{5}{\sqrt{50}} = \frac{5}{5\sqrt{2}} = \frac{1}{\sqrt{2}} \]

### Step 6: Find the angle \(\alpha\)
To find \(\alpha\), we take the inverse cosine:
\[ \alpha = \cos^{-1}\left(\frac{1}{\sqrt{2}}\right) \]
This gives:
\[ \alpha = 45^\circ \]

### Final Answer
The angle made by \(\vec{C}\) with the x-axis is \(45^\circ\).

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