The angle between the vectors `(i + j + k) and (i- j - k)` is

Answer

To find the angle between the vectors \( \mathbf{a} = \mathbf{i} + \mathbf{j} + \mathbf{k} \) and \( \mathbf{b} = \mathbf{i} - \mathbf{j} - \mathbf{k} \), we can use the formula for the cosine of the angle \( \theta \) between two vectors:

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

### Step 1: Calculate the dot product \( \mathbf{a} \cdot \mathbf{b} \)

The dot product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is calculated as follows:

\[ \mathbf{a} \cdot \mathbf{b} = (1)(1) + (1)(-1) + (1)(-1) \]

Calculating this gives:

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

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

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

\[ |\mathbf{a}| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3} \]

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

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

### Step 3: Substitute into the cosine formula

Now we can substitute the values into the cosine formula:

\[ \cos \theta = \frac{-1}{\sqrt{3} \cdot \sqrt{3}} = \frac{-1}{3} \]

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

To find \( \theta \), we take the inverse cosine:

\[ \theta = \cos^{-1}\left(-\frac{1}{3}\right) \]

This gives us the angle between the two vectors.

### Final Answer

Thus, the angle between the vectors \( \mathbf{a} \) and \( \mathbf{b} \) is:

\[ \theta = \cos^{-1}\left(-\frac{1}{3}\right) \]

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