What is the vector perpendicular to both the vectors `hat(i)-hat(j) and hat(i)` ?

Options

A. `hat(i)`
B. `-hat(j)`
C. `hat(j)`
D. `hat(k)`

Explanation

To find a vector that is perpendicular to both the vectors \(\hat{i} - \hat{j}\) and \(\hat{i}\), we can use the concept of the dot product. A vector \(\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}\) is perpendicular to another vector \(\vec{a}\) if their dot product is zero, i.e., \(\vec{r} \cdot \vec{a} = 0\).

### Step 1: Set up the equations for perpendicularity
We need to find a vector \(\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}\) that is perpendicular to both \(\hat{i} - \hat{j}\) and \(\hat{i}\).

1. **First condition**: \(\vec{r} \cdot (\hat{i} - \hat{j}) = 0\)
\[ (x\hat{i} + y\hat{j} + z\hat{k}) \cdot (\hat{i} - \hat{j}) = 0 \]
This simplifies to:
\[ x(1) + y(-1) + z(0) = 0 \implies x - y = 0 \implies x = y \]

2. **Second condition**: \(\vec{r} \cdot \hat{i} = 0\)
\[ (x\hat{i} + y\hat{j} + z\hat{k}) \cdot \hat{i} = 0 \]
This simplifies to:
\[ x(1) + y(0) + z(0) = 0 \implies x = 0 \]

### Step 2: Solve the equations
From the first condition, we have \(x = y\). From the second condition, we have \(x = 0\).

Substituting \(x = 0\) into \(x = y\) gives us:
\[ y = 0 \]

### Step 3: Determine the value of z
Since \(x\) and \(y\) are both 0, the vector \(\vec{r}\) can be expressed as:
\[ \vec{r} = 0\hat{i} + 0\hat{j} + z\hat{k} = z\hat{k} \]
Here, \(z\) can be any non-zero value. Therefore, a vector perpendicular to both \(\hat{i} - \hat{j}\) and \(\hat{i}\) is:
\[ \vec{r} = k\hat{k} \quad \text{(where \(k\) is a non-zero scalar)} \]

### Conclusion
The vector perpendicular to both \(\hat{i} - \hat{j}\) and \(\hat{i}\) is any scalar multiple of \(\hat{k}\).