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 solve the problem, we need to find the angle between the vector \( \mathbf{R} = \mathbf{A} + \mathbf{B} \) and the vector \( \mathbf{P} = \hat{i} + 2\hat{j} - 4\hat{k} \).

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

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 vectors:
\[ \mathbf{R} = \mathbf{A} + \mathbf{B} = (-4\hat{i} + 4\hat{j} + 2\hat{k}) + (2\hat{i} - \hat{j} - \hat{k}) \]
Combine like terms:
\[ \mathbf{R} = (-4 + 2)\hat{i} + (4 - 1)\hat{j} + (2 - 1)\hat{k} = -2\hat{i} + 3\hat{j} + 1\hat{k} \]

### Step 2: Calculate the dot product \( \mathbf{P} \cdot \mathbf{R} \)

Now, we need to find the dot product \( \mathbf{P} \cdot \mathbf{R} \):
\[ \mathbf{P} = \hat{i} + 2\hat{j} - 4\hat{k} \]
\[ \mathbf{R} = -2\hat{i} + 3\hat{j} + 1\hat{k} \]

The dot product is calculated as:
\[ \mathbf{P} \cdot \mathbf{R} = (1)(-2) + (2)(3) + (-4)(1) \]
Calculating each term:
\[ = -2 + 6 - 4 = 0 \]

### Step 3: Calculate the magnitudes of \( \mathbf{P} \) and \( \mathbf{R} \)

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

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

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

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

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

Since \( \cos(\theta) = 0 \), this implies:
\[ \theta = 90^\circ \]

Thus, the angle made by \( \mathbf{A} + \mathbf{B} \) with \( \hat{i} + 2\hat{j} - 4\hat{k} \) is \( 90^\circ \).

### Final Answer
The angle is \( 90^\circ \).

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