Exercise · ZW-LINES-AND-PLANES-IN-SPACE-001 · ★☆☆ Basic
What is the equation of the plane through \((1,2,3)\) with normal vector \(\langle2,-1,4\rangle\)?
Choose one option.
Show answer
- C. \(2x-y+4z=12\)
Explanation
The equation is \(\mathbf n\cdot(\mathbf r-\mathbf r_0)=0\), i.e. \(2(x-1)-(y-2)+4(z-3)=0\). Simplifying gives \(2x-y+4z=2-2+12=12\).
Review the concept: Lines and planes in space
A line in space is determined by a point \(\mathbf r_0\) and a direction vector \(\mathbf d\); a plane, by a point \(\mathbf r_0\) and a normal vector \(\mathbf n\) perpendicular to it.
\[\begin{aligned}\mathbf r(t)&=\mathbf r_0+t\,\mathbf d\\\mathbf n\cdot(\mathbf r-\mathbf r_0)&=0\end{aligned}\]
More practice problems: lines and planes in space
- ZW-LINES-AND-PLANES-IN-SPACE-002 A line passes through (1,0,2) with direction langle3,-1,4rangle. What are its parametric equations?
- ZW-LINES-AND-PLANES-IN-SPACE-003 What is a direction vector of the line through (1,2,3) and (4,0,5)?
- ZW-LINES-AND-PLANES-IN-SPACE-004 What is the equation of the plane through (1,0,0), (0,1,0) and (0,0,1)?
- ZW-LINES-AND-PLANES-IN-SPACE-005 Which of the following points lies on the plane x+2y-z=4?
- ZW-LINES-AND-PLANES-IN-SPACE-006 Which of the planes is parallel to x-2y+3z=5?
- ZW-LINES-AND-PLANES-IN-SPACE-007 At which point does the line (1+t, 2-t, 3+2t) meet the plane x+y+z=10?
- ZW-LINES-AND-PLANES-IN-SPACE-008 What is the relation between the line (1,1,1)+tlangle2,-1,3rangle and the plane 4x-2y+6z=5?
- ZW-LINES-AND-PLANES-IN-SPACE-009 What is the equation of the plane through the origin containing the directions langle1,2,0rangle an…