Given two vectors `vec(A) = -hat(i) + 2hat(j) - 3hat(k)` and `vec(B) = 4hat(i) - 2hat(j) + 6hat(k)`. The angle made by (A+B) with x-axis is :

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. `90^(@)`

Answer

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

### Step 1: Calculate \( \vec{A} + \vec{B} \)

Given:
\[ \vec{A} = -\hat{i} + 2\hat{j} - 3\hat{k} \]
\[ \vec{B} = 4\hat{i} - 2\hat{j} + 6\hat{k} \]

Now, we add the vectors \( \vec{A} \) and \( \vec{B} \):
\[ \vec{A} + \vec{B} = (-\hat{i} + 4\hat{i}) + (2\hat{j} - 2\hat{j}) + (-3\hat{k} + 6\hat{k}) \]
Calculating each component:
- For \( \hat{i} \): \( -1 + 4 = 3 \)
- For \( \hat{j} \): \( 2 - 2 = 0 \)
- For \( \hat{k} \): \( -3 + 6 = 3 \)

Thus,
\[ \vec{A} + \vec{B} = 3\hat{i} + 0\hat{j} + 3\hat{k} \]

### Step 2: Find the angle with the x-axis

To find the angle \( \theta \) that the vector \( \vec{A} + \vec{B} \) makes with the x-axis, we use the cosine of the angle:
\[ \cos \theta = \frac{\text{Component along } x}{\text{Magnitude of the vector}} \]

The component along the x-axis is 3 (from \( 3\hat{i} \)).

### Step 3: Calculate the magnitude of \( \vec{A} + \vec{B} \)

The magnitude of \( \vec{A} + \vec{B} \) is given by:
\[ |\vec{A} + \vec{B}| = \sqrt{(3)^2 + (0)^2 + (3)^2} = \sqrt{9 + 0 + 9} = \sqrt{18} = 3\sqrt{2} \]

### Step 4: Calculate \( \cos \theta \)

Now substituting the values into the cosine formula:
\[ \cos \theta = \frac{3}{3\sqrt{2}} = \frac{1}{\sqrt{2}} \]

### Step 5: Find \( \theta \)

To find \( \theta \):
\[ \theta = \cos^{-1}\left(\frac{1}{\sqrt{2}}\right) \]
This corresponds to:
\[ \theta = 45^\circ \]

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

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