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

Options

A. `(pi)/(6)`
B. `(pi)/(4)`
C. `(pi)/(3)`
D. `(2pi)/(3)`

Answer

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

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

where \(\vec{a}\) and \(\vec{b}\) are the vectors, \(\vec{a} \cdot \vec{b}\) is the dot product of the vectors, and \(|\vec{a}|\) and \(|\vec{b}|\) are the magnitudes of the vectors.

### Step 1: Identify the vectors
Let:
\[ \vec{a} = \hat{i} - \hat{j} \]
\[ \vec{b} = \hat{j} + \hat{k} \]

### Step 2: Calculate the dot product \(\vec{a} \cdot \vec{b}\)
The dot product is calculated as follows:
\[ \vec{a} \cdot \vec{b} = (\hat{i} - \hat{j}) \cdot (\hat{j} + \hat{k}) \]
Using the distributive property:
\[ = \hat{i} \cdot \hat{j} + \hat{i} \cdot \hat{k} - \hat{j} \cdot \hat{j} - \hat{j} \cdot \hat{k} \]
Since \(\hat{i} \cdot \hat{j} = 0\), \(\hat{i} \cdot \hat{k} = 0\), \(\hat{j} \cdot \hat{j} = 1\), and \(\hat{j} \cdot \hat{k} = 0\):
\[ = 0 + 0 - 1 - 0 = -1 \]

### Step 3: Calculate the magnitudes of \(\vec{a}\) and \(\vec{b}\)
The magnitude of \(\vec{a}\) is:
\[ |\vec{a}| = \sqrt{(\hat{i} - \hat{j}) \cdot (\hat{i} - \hat{j})} = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

The magnitude of \(\vec{b}\) is:
\[ |\vec{b}| = \sqrt{(\hat{j} + \hat{k}) \cdot (\hat{j} + \hat{k})} = \sqrt{1^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 4: Substitute into the cosine formula
Now substituting the values into the cosine formula:
\[ \cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|} = \frac{-1}{\sqrt{2} \cdot \sqrt{2}} = \frac{-1}{2} \]

### Step 5: Find the angle \(\theta\)
To find \(\theta\), we take the inverse cosine:
\[ \theta = \cos^{-1}(-\frac{1}{2}) \]
The angle whose cosine is \(-\frac{1}{2}\) is:
\[ \theta = \frac{2\pi}{3} \text{ (or } 120^\circ \text{)} \]

### Final Answer
Thus, the angle between the vectors \(\hat{i} - \hat{j}\) and \(\hat{j} + \hat{k}\) is:
\[ \theta = \frac{2\pi}{3} \]