3D Geometry
Vector Equation of Line
Grade 12
Question:
<p>The vector equation of the straight line passing through \((1, 2, 3)\) and perpendicular to the plane \(\vec{r} \cdot (\vec{i} + 2\vec{j} - 5\vec{k}) + 9 = 0\) is</p>
<p>(a) \(\vec{r} = (\vec{i} + 2\vec{j} + 3\vec{k}) + \lambda(\vec{i} + 2\vec{j} - 5\vec{k})\)</p>
<p>(b) \(\vec{r} = (\vec{i} + 2\vec{j} - 5\vec{k}) + \lambda(\vec{i} + 2\vec{j} + 3\vec{k})\)</p>
<p>(c) \(\vec{r} = (\vec{i} + 2\vec{j} + 3\vec{k}) + \lambda(-8\vec{k})\)</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: The direction vector of a line perpendicular to a plane is the normal vector of that plane.
A line perpendicular to a plane has its direction vector parallel to the normal vector of the plane. The normal vector is $(\hat{i} + 2\hat{j} - 5\hat{k})$.
Correct Answer: A