Given two vectors `A = -4hat(i) + 4hat(j) + 2hat(k)` and `B = 2hat(i) - hat(j) - hat(k)`. The angle made by (A+B) with `hat(i) + 2hat(j) - 4hat(k)` 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 vector \( \mathbf{X} = \hat{i} + 2\hat{j} - 4\hat{k} \), we will follow these steps:

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

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

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

### Step 2: Calculate the dot product \( (\mathbf{A} + \mathbf{B}) \cdot \mathbf{X} \)

Given:
\[ \mathbf{X} = \hat{i} + 2\hat{j} - 4\hat{k} \]

Now, we calculate the dot product:
\[ (\mathbf{A} + \mathbf{B}) \cdot \mathbf{X} = (-2\hat{i} + 3\hat{j} + \hat{k}) \cdot (\hat{i} + 2\hat{j} - 4\hat{k}) \]
\[ = (-2)(1) + (3)(2) + (1)(-4) \]
\[ = -2 + 6 - 4 \]
\[ = 0 \]

### Step 3: Calculate the magnitudes of \( \mathbf{A} + \mathbf{B} \) and \( \mathbf{X} \)

Magnitude of \( \mathbf{A} + \mathbf{B} \):
\[ |\mathbf{A} + \mathbf{B}| = \sqrt{(-2)^2 + 3^2 + 1^2} = \sqrt{4 + 9 + 1} = \sqrt{14} \]

Magnitude of \( \mathbf{X} \):
\[ |\mathbf{X}| = \sqrt{(1)^2 + (2)^2 + (-4)^2} = \sqrt{1 + 4 + 16} = \sqrt{21} \]

### Step 4: Calculate the angle \( \theta \)

Using the formula for the cosine of the angle between two vectors:
\[ \cos \theta = \frac{(\mathbf{A} + \mathbf{B}) \cdot \mathbf{X}}{|\mathbf{A} + \mathbf{B}| |\mathbf{X}|} \]
Substituting the values we found:
\[ \cos \theta = \frac{0}{\sqrt{14} \cdot \sqrt{21}} = 0 \]

### Step 5: Determine the angle \( \theta \)

Since \( \cos \theta = 0 \), we find:
\[ \theta = 90^\circ \]

### Final Answer:
The angle made by \( \mathbf{A} + \mathbf{B} \) with \( \hat{i} + 2\hat{j} - 4\hat{k} \) is \( 90^\circ \).

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