If `vec(r )=x hat(i)+y hat(j)+x hat(k)`, find : `(vec(r )xx hat(i)).(vec(r )xx hat(j))+xy`.

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem, we need to calculate \( \vec{r} \times \hat{i} \cdot \vec{r} \times \hat{j} + xy \). Let's break this down step by step.

### Step 1: Define the vector \(\vec{r}\)
Given:
\[ \vec{r} = x \hat{i} + y \hat{j} + x \hat{k} \]

### Step 2: Calculate \(\vec{r} \times \hat{i}\)
Using the cross product formula:
\[ \vec{r} \times \hat{i} = (x \hat{i} + y \hat{j} + x \hat{k}) \times \hat{i} \]
Using the properties of the cross product:
- \(\hat{i} \times \hat{i} = \vec{0}\)
- \(\hat{j} \times \hat{i} = -\hat{k}\)
- \(\hat{k} \times \hat{i} = \hat{j}\)

Calculating:
\[ \vec{r} \times \hat{i} = (x \hat{i}) \times \hat{i} + (y \hat{j}) \times \hat{i} + (x \hat{k}) \times \hat{i} \]
\[ = \vec{0} + y (-\hat{k}) + x \hat{j} \]
\[ = -y \hat{k} + x \hat{j} \]
Thus,
\[ \vec{r} \times \hat{i} = x \hat{j} - y \hat{k} \]

### Step 3: Calculate \(\vec{r} \times \hat{j}\)
Now, calculate:
\[ \vec{r} \times \hat{j} = (x \hat{i} + y \hat{j} + x \hat{k}) \times \hat{j} \]
Using the properties of the cross product:
- \(\hat{j} \times \hat{j} = \vec{0}\)
- \(\hat{i} \times \hat{j} = \hat{k}\)
- \(\hat{k} \times \hat{j} = -\hat{i}\)

Calculating:
\[ \vec{r} \times \hat{j} = (x \hat{i}) \times \hat{j} + (y \hat{j}) \times \hat{j} + (x \hat{k}) \times \hat{j} \]
\[ = x \hat{k} + \vec{0} + x (-\hat{i}) \]
\[ = x \hat{k} - x \hat{i} \]
Thus,
\[ \vec{r} \times \hat{j} = -x \hat{i} + x \hat{k} \]

### Step 4: Calculate the dot product \((\vec{r} \times \hat{i}) \cdot (\vec{r} \times \hat{j})\)
Now we find:
\[ (\vec{r} \times \hat{i}) \cdot (\vec{r} \times \hat{j}) = (x \hat{j} - y \hat{k}) \cdot (-x \hat{i} + x \hat{k}) \]
Calculating the dot product:
\[ = (x \hat{j}) \cdot (-x \hat{i}) + (x \hat{j}) \cdot (x \hat{k}) + (-y \hat{k}) \cdot (-x \hat{i}) + (-y \hat{k}) \cdot (x \hat{k}) \]
\[ = 0 + 0 + 0 - yx \]
Thus,
\[ (\vec{r} \times \hat{i}) \cdot (\vec{r} \times \hat{j}) = -xy \]

### Step 5: Combine with \(xy\)
Now we add \(xy\):
\[ (\vec{r} \times \hat{i}) \cdot (\vec{r} \times \hat{j}) + xy = -xy + xy = 0 \]

### Final Answer
The final result is:
\[ \boxed{0} \]