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: Plan:
Use the direction ratios of the Y-axis.

Step 2: Steps:
A line perpendicular to the XZ-plane is parallel to the Y-axis, so its direction ratios are $0, 1, 0$. With the given point, the vector form uses $\lambda\hat j$ as the direction term. Only option C has this.

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