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

Options

A. `sin^(-1)((sqrt(8))/3)`
B. `sin^(-1)(1/3)+(pi)/3`
C. `cos^(-1)((sqrt(8))/3)`
D. `cos^(-1)sqrt(8/3)`

Explanation

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

\[ \cos \theta = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|} \]

where \( \vec{A} = \hat{i} + \hat{j} + \hat{k} \) and \( \vec{B} = \hat{i} - \hat{j} - \hat{k} \).

### Step 1: Calculate the dot product \( \vec{A} \cdot \vec{B} \)

The dot product is calculated as follows:

\[ \vec{A} \cdot \vec{B} = (\hat{i} + \hat{j} + \hat{k}) \cdot (\hat{i} - \hat{j} - \hat{k}) \]

Calculating this gives:

\[ = \hat{i} \cdot \hat{i} + \hat{j} \cdot (-\hat{j}) + \hat{k} \cdot (-\hat{k}) \]
\[ = 1 - 1 - 1 = -1 \]

### Step 2: Calculate the magnitudes \( |\vec{A}| \) and \( |\vec{B}| \)

First, calculate \( |\vec{A}| \):

\[ |\vec{A}| = \sqrt{(1^2 + 1^2 + 1^2)} = \sqrt{3} \]

Now, calculate \( |\vec{B}| \):

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

### Step 3: Substitute into the cosine formula

Now we substitute the dot product and magnitudes into the cosine formula:

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

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

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

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

### Step 5: Express in terms of sine (optional)

If we want to express it in terms of sine, we can use the identity:

\[ \sin^2 \theta + \cos^2 \theta = 1 \]

Calculating \( \sin^2 \theta \):

\[ \sin^2 \theta = 1 - \left(-\frac{1}{3}\right)^2 = 1 - \frac{1}{9} = \frac{8}{9} \]

Thus,

\[ \sin \theta = \sqrt{\frac{8}{9}} = \frac{\sqrt{8}}{3} \]

So we can also express the angle as:

\[ \theta = \sin^{-1}\left(\frac{\sqrt{8}}{3}\right) \]

### Final Answer

The angle between the vectors \( \hat{i} + \hat{j} + \hat{k} \) and \( \hat{i} - \hat{j} - \hat{k} \) is:

\[ \theta = \cos^{-1}\left(-\frac{1}{3}\right) \quad \text{or} \quad \sin^{-1}\left(\frac{\sqrt{8}}{3}\right) \]