Question

The equation of a line passing through a point \((4,-2,3)\) and perpendicular to the XZ-plane is....

• A. \(\overset{̄}{r} = (4\hat{i}-2\hat{j}+3\hat{k})+λ(\hat{i}+\hat{k})\)
• B. \(\overset{̄}{r} = (4\hat{i}-2\hat{j}+3\hat{k})+λ(\hat{i})\)
• C. \(\overset{̄}{r} = (4\hat{i}-2\hat{j}+3\hat{k})+λ(\hat{j})\)
• D. \(\overset{̄}{r} = (4\hat{i}-2\hat{j}+3\hat{k})+λ(\hat{i}-\hat{k})\)

Hint:

The XZ-plane has the Y-axis as its normal, so the line runs along \(\hat j\).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
A line perpendicular to a plane runs along the normal to that plane.

Step 2: Find the direction:
The XZ-plane is the plane \(y = 0\). Its normal is along the Y-axis, i.e. along \(\hat j\).

Step 3: Write the line:
Through \((4,-2,3)\) with direction \(\hat j\): \(\vec r = (4\hat i - 2\hat j + 3\hat k) + \lambda\hat j\).
Options A and D have components along \(\hat i\) and \(\hat k\), which lie in the XZ-plane. Option B runs along the X-axis, which is parallel to the plane.

Final Answer:
The line is \(\vec r = (4\hat i-2\hat j+3\hat k)+\lambda\hat j\), option (C). \[ \boxed{\vec r = (4\hat i-2\hat j+3\hat k)+\lambda\hat j} \]