What is the interior acute angle of the parallelogram whose sides are represented by the vectors `(1)/(sqrt2) hati + (1)/(sqrt2) hatj + hatk` and `(1)/(sqrt2) hati - (1)/(sqrt2) hatj +hatk` ?

Options

A. `60^@`
B. `45^@`
C. `30^@`
D. `15^@`

Answer

To find the interior acute angle of the parallelogram whose sides are represented by the vectors \( \mathbf{a} = \frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j} + \hat{k} \) and \( \mathbf{b} = \frac{1}{\sqrt{2}} \hat{i} - \frac{1}{\sqrt{2}} \hat{j} + \hat{k} \), we can follow these steps:

### Step 1: Calculate the dot product of the vectors \( \mathbf{a} \) and \( \mathbf{b} \).

The dot product \( \mathbf{a} \cdot \mathbf{b} \) is calculated as follows:

\[ \mathbf{a} \cdot \mathbf{b} = \left( \frac{1}{\sqrt{2}} \right) \left( \frac{1}{\sqrt{2}} \right) + \left( \frac{1}{\sqrt{2}} \right) \left( -\frac{1}{\sqrt{2}} \right) + (1)(1) \]

Calculating each term:

\[ \mathbf{a} \cdot \mathbf{b} = \frac{1}{2} - \frac{1}{2} + 1 = 1 \]

### Step 2: Calculate the magnitudes of the vectors \( \mathbf{a} \) and \( \mathbf{b} \).

The magnitude of vector \( \mathbf{a} \) is given by:

\[ |\mathbf{a}| = \sqrt{\left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 + (1)^2} = \sqrt{\frac{1}{2} + \frac{1}{2} + 1} = \sqrt{2} \]

The magnitude of vector \( \mathbf{b} \) is calculated similarly:

\[ |\mathbf{b}| = \sqrt{\left(\frac{1}{\sqrt{2}}\right)^2 + \left(-\frac{1}{\sqrt{2}}\right)^2 + (1)^2} = \sqrt{\frac{1}{2} + \frac{1}{2} + 1} = \sqrt{2} \]

### Step 3: Use the dot product to find the cosine of the angle \( \theta \).

Using the formula for the dot product:

\[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta \]

Substituting the values we found:

\[ 1 = (\sqrt{2})(\sqrt{2}) \cos \theta \]

This simplifies to:

\[ 1 = 2 \cos \theta \]

### Step 4: Solve for \( \cos \theta \).

\[ \cos \theta = \frac{1}{2} \]

### Step 5: Find the angle \( \theta \).

The angle \( \theta \) whose cosine is \( \frac{1}{2} \) is:

\[ \theta = 60^\circ \]

### Conclusion:

The interior acute angle of the parallelogram is \( 60^\circ \).

---