Step 1: Understanding the Concept:
To find the intercept of a plane on a coordinate axis (say the X-axis), we set the position vector \( \vec{r} \) to represent a point on that axis. A point on the X-axis has the form \( \vec{r} = x\hat{i} \).
Step 2: Key Formula or Approach:
1. For X-intercept: Let \( \vec{r} = x\hat{i} \) and solve for \( x \).
2. For Y-intercept: Let \( \vec{r} = y\hat{j} \) and solve for \( y \).
3. For Z-intercept: Let \( \vec{r} = z\hat{k} \) and solve for \( z \).
Step 3: Detailed Explanation:
Substitute \( \vec{r} = x\hat{i} \) into the plane equation \( \vec{r} \cdot \vec{n} = d \):
\[ (x\hat{i}) \cdot \vec{n} = d \implies x(\hat{i} \cdot \vec{n}) = d \implies x = \frac{d}{\hat{i} \cdot \vec{n}} \]
Similarly, for the Y-intercept, let \( \vec{r} = y\hat{j} \):
\[ (y\hat{j}) \cdot \vec{n} = d \implies y(\hat{j} \cdot \vec{n}) = d \implies y = \frac{d}{\hat{j} \cdot \vec{n}} \]
And for the Z-intercept, let \( \vec{r} = z\hat{k} \):
\[ (z\hat{k}) \cdot \vec{n} = d \implies z(\hat{k} \cdot \vec{n}) = d \implies z = \frac{d}{\hat{k} \cdot \vec{n}} \]
Step 4: Final Answer:
The intercepts are \( \frac{d}{\hat{i} \cdot \vec{n}}, \frac{d}{\hat{j} \cdot \vec{n}}, \frac{d}{\hat{k} \cdot \vec{n}} \), which is option (C).