Find the angle between `hat(i)+hat(j)+hat(k)` and `hat(i)+hat(j)-hat(k)`.

Answer

To find the angle between the vectors \( \hat{i} + \hat{j} + \hat{k} \) and \( \hat{i} + \hat{j} - \hat{k} \), we can follow these steps:

### Step 1: Define the Vectors
Let:
- \( \mathbf{a} = \hat{i} + \hat{j} + \hat{k} \)
- \( \mathbf{b} = \hat{i} + \hat{j} - \hat{k} \)

### Step 2: Calculate the Dot Product
The dot product \( \mathbf{a} \cdot \mathbf{b} \) is calculated as follows:
\[ \mathbf{a} \cdot \mathbf{b} = (1)(1) + (1)(1) + (1)(-1) = 1 + 1 - 1 = 1 \]

### Step 3: Calculate the Magnitudes of the Vectors
To find the magnitudes of \( \mathbf{a} \) and \( \mathbf{b} \):

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

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

### Step 4: Use the Dot Product to Find the Cosine of the Angle
Using the formula for the dot product:
\[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta \]
Substituting the values we have:
\[ 1 = \sqrt{3} \cdot \sqrt{3} \cdot \cos \theta \]
\[ 1 = 3 \cos \theta \]
\[ \cos \theta = \frac{1}{3} \]

### Step 5: Find the Angle
To find the angle \( \theta \):
\[ \theta = \cos^{-1}\left(\frac{1}{3}\right) \]

### Final Answer
The angle between the vectors \( \hat{i} + \hat{j} + \hat{k} \) and \( \hat{i} + \hat{j} - \hat{k} \) is:
\[ \theta = \cos^{-1}\left(\frac{1}{3}\right) \]