Points, directions, and normals in space

Mathematics · Geometry · ages 19-20

Name ______________________ Date ____________

The line r = (1, 0, 2) + t(3, -1, 2) passes through (1, 0, 2) when t = 0. What is its x-coordinate when t = 2?
Answer: ______________
Which vector is normal to the plane 2x - y + 4z = 7?
- (2, -1, 4)
- (2, 1, 4)
- (7, -1, 4)
- (2, -1, 7)
Which vector is normal to the plane 2x minus y plus 4z = 7?
- (2, 1, 4)
- (2, negative 1, 4)
- (7, negative 1, 4)
What is the distance from the point (0, 0, 5) to the plane z = 2? Give a number.
Answer: ______________
What is the angle between the coordinate planes x = 0 and y = 0?
- 45 degrees
- 90 degrees
- 0 degrees
- 60 degrees
A plane has normal vector (1, 2, negative 1) and passes through (3, 0, 1). Written as x plus 2y minus z = k, what is k? Give a number.
Answer: ______________
Line L1 passes through (0, 0, 0) with direction (1, 0, 0). Line L2 passes through (0, 1, 0) with direction (0, 0, 1). How are the lines related?
- Parallel
- Intersecting
- Skew
Line L1 passes through (0, 0, 0) with direction (1, 0, 0). Line L2 passes through (0, 1, 0) with direction (0, 0, 1). How are the lines related?
- parallel
- intersecting
- skew
- identical
Two lines are skew. Which operation starts the distance calculation between them?
- The cross product of the two direction vectors
- The dot product of the two direction vectors
- The sum of the two direction vectors
A plane has normal vector (1, 2, -1) and passes through (3, 0, 1). Written in the form x + 2y - z = k, what is k?
Answer: ______________

Answer key

For grown-ups. Fold this page away before handing over the rest.

7 · x = 1 + 3t, so at t = 2 the x-coordinate is 1 + 6 = 7.
(2, -1, 4) · The coefficients of x, y and z in the plane equation give a normal vector directly: (2, -1, 4).
(2, negative 1, 4) · The coefficients of x, y and z in the plane equation give a normal vector directly.
3 · The plane z = 2 is horizontal, so the distance is the vertical gap: 5 minus 2.
90 degrees · The normals (1, 0, 0) and (0, 1, 0) have dot product 0, so the planes are perpendicular: 90 degrees.
2 · Substitute the point: 3 plus 2 times 0 minus 1 is 2, so k is 2.
Skew · The directions are not multiples, so the lines are not parallel, yet L1 has y = 0 while L2 has y = 1, so they never meet.
skew · The directions are not multiples, so the lines are not parallel. L1 has y = 0, z = 0 while L2 has y = 1, so they never meet. Nonparallel lines that never meet are skew.
The cross product of the two direction vectors · The common perpendicular comes from the cross product, and its length over the direction lengths gives the gap.
2 · Substitute the point: 3 + 2 x 0 - 1 = 2, so k = 2.