Calculus Volume 3 · free preview
Equations of Lines and Planes in Space
Lines in space need a point and a direction vector, planes need a point and a normal vector, with parametric, symmetric, and point-normal forms the working equations and distances from points to lines and planes following from projections.
1. Lines in space
Lines in space
2. Planes and the normal vector
Planes and the normal vector
3. Distances
Distances
4. Intersections and angles
Intersections and angles
5. Line direction vector
The parametric line x = 1 + 2t, y = 3 - t, z = 2 + 4t has which direction vector?
6. Plane normal from points
To find the plane through three points, the normal vector is obtained how?
7. Point-normal form
The plane with normal (1, 2, 3) through (0, 0, 0) has which equation?
8. Point to plane distance
The distance from (x_1, y_1, z_1) to the plane Ax + By + Cz = D is which expression?
9. Line-plane intersection
To find where a parametric line meets a plane, what does one do?
10. Symmetric equations denominator
The symmetric equations of a line divide x - x_0 by a. As a single letter, what is a? Answer with one lowercase letter.
11. Skew lines count
Two nonparallel nonintersecting lines in space are called skew. As a number, how many such lines can exist in a plane? Answer with a number.
12. Plane dimension match
A plane's general form is one scalar equation, matching its how many intrinsic dimensions? Answer with a number.
13. Plane-plane angle
The angle between two intersecting planes is defined as the angle between which vectors?
14. Normal magnitude in distance
The point-to-plane distance divides by the root of A squared + B squared + C squared, the magnitude of what vector? Answer with two words.
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