Given two vectors A = i-2j-3k and B = 4i - 2j + 6k. The angle made by (A + B) with the X-axis is

Options

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

Answer

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

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

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

Now, add the two vectors:
\[ \mathbf{A} + \mathbf{B} = (\mathbf{i} + 4\mathbf{i}) + (-2\mathbf{j} - 2\mathbf{j}) + (-3\mathbf{k} + 6\mathbf{k}) \]
\[ = (1 + 4)\mathbf{i} + (-2 - 2)\mathbf{j} + (-3 + 6)\mathbf{k} \]
\[ = 5\mathbf{i} - 4\mathbf{j} + 3\mathbf{k} \]

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

The angle \( \theta \) that the vector \( \mathbf{R} = \mathbf{A} + \mathbf{B} \) makes with the X-axis can be found using the dot product formula. The dot product of two vectors \( \mathbf{X} \) (along the X-axis) and \( \mathbf{R} \) is given by:
\[ \mathbf{X} \cdot \mathbf{R} = |\mathbf{X}| |\mathbf{R}| \cos \theta \]

Where:
\[ \mathbf{X} = \mathbf{i} \quad \text{(which has components (1, 0, 0))} \]
\[ \mathbf{R} = 5\mathbf{i} - 4\mathbf{j} + 3\mathbf{k} \]

### Step 3: Calculate the magnitudes

1. **Magnitude of \( \mathbf{R} \)**:
\[ |\mathbf{R}| = \sqrt{(5)^2 + (-4)^2 + (3)^2} = \sqrt{25 + 16 + 9} = \sqrt{50} = 5\sqrt{2} \]

2. **Magnitude of \( \mathbf{X} \)**:
\[ |\mathbf{X}| = 1 \]

### Step 4: Calculate the dot product

Now, calculate the dot product \( \mathbf{X} \cdot \mathbf{R} \):
\[ \mathbf{X} \cdot \mathbf{R} = (1)(5) + (0)(-4) + (0)(3) = 5 \]

### Step 5: Set up the equation

Using the dot product formula:
\[ 5 = 1 \cdot (5\sqrt{2}) \cos \theta \]
\[ 5 = 5\sqrt{2} \cos \theta \]

### Step 6: Solve for \( \cos \theta \)

Rearranging gives:
\[ \cos \theta = \frac{5}{5\sqrt{2}} = \frac{1}{\sqrt{2}} \]

### Step 7: Find \( \theta \)

Now, find \( \theta \):
\[ \theta = \cos^{-1}\left(\frac{1}{\sqrt{2}}\right) = 45^\circ \]

### Final Answer

The angle made by \( \mathbf{A} + \mathbf{B} \) with the X-axis is \( 45^\circ \).

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