Exercise · ZW-LINES-AND-PLANES-IN-SPACE-004 · ★★☆ Intermediate
What is the equation of the plane through \((1,0,0)\), \((0,1,0)\) and \((0,0,1)\)?
Choose one option.
Show answer
- B. \(x+y+z=1\)
Explanation
Two vectors in the plane are \(\langle-1,1,0\rangle\) and \(\langle-1,0,1\rangle\); their cross product is \(\langle1,1,1\rangle\), the normal. With the point \((1,0,0)\) we get \(x+y+z=1\).
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-001 What is the equation of the plane through (1,2,3) with normal vector langle2,-1,4rangle?
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- 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-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…
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