If `vec(r)=xhat(i)+yhat(j)+zhat(k)`, then what is `vec(r).(hat(i)+hat(j)+hat(k))` equal to?

Options

A. x
B. x+y
C. `-(x+y+z)`
D. `(x+y+z)`

Explanation

To solve the problem, we need to calculate the dot product of the vector \(\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}\) with the vector \((\hat{i} + \hat{j} + \hat{k})\).

### Step-by-Step Solution:

1. **Identify the vectors**:
- The vector \(\vec{r}\) is given as:
\[ \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} \]
- The vector we are taking the dot product with is:
\[ \vec{a} = \hat{i} + \hat{j} + \hat{k} \]

2. **Write the dot product**:
- We need to compute \(\vec{r} \cdot \vec{a}\):
\[ \vec{r} \cdot \vec{a} = (x\hat{i} + y\hat{j} + z\hat{k}) \cdot (\hat{i} + \hat{j} + \hat{k}) \]

3. **Apply the distributive property of the dot product**:
- Using the distributive property, we can expand the dot product:
\[ \vec{r} \cdot \vec{a} = x\hat{i} \cdot \hat{i} + x\hat{i} \cdot \hat{j} + x\hat{i} \cdot \hat{k} + y\hat{j} \cdot \hat{i} + y\hat{j} \cdot \hat{j} + y\hat{j} \cdot \hat{k} + z\hat{k} \cdot \hat{i} + z\hat{k} \cdot \hat{j} + z\hat{k} \cdot \hat{k} \]

4. **Evaluate each dot product**:
- Recall that the dot product of two different unit vectors is zero, and the dot product of a unit vector with itself is one:
- \(\hat{i} \cdot \hat{i} = 1\)
- \(\hat{j} \cdot \hat{j} = 1\)
- \(\hat{k} \cdot \hat{k} = 1\)
- All other combinations (e.g., \(\hat{i} \cdot \hat{j}\), \(\hat{i} \cdot \hat{k}\), etc.) are zero.
- Thus, we have:
\[ \vec{r} \cdot \vec{a} = x(1) + y(1) + z(1) + 0 + 0 + 0 + 0 + 0 + 0 = x + y + z \]

5. **Final result**:
- Therefore, the result of the dot product is:
\[ \vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = x + y + z \]

### Conclusion:
The answer is \(x + y + z\).

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