Calculus Volume 3 · free preview
Vectors in Three Dimensions
Three-dimensional space adds a third coordinate and the basis vector k, with distance, spheres, and the geometry of space built on ordered triples and the same component operations as the plane.
1. The third coordinate
The third coordinate
2. Distance and spheres
Distance and spheres
3. The standard basis in space
The standard basis in space
4. Projections and shadows
Projections and shadows
5. Third basis vector
In three dimensions the standard basis adds which vector to i and j?
6. Space distance
The distance from (0, 0, 0) to (1, 2, 2) equals what number?
7. Sphere equation
The sphere centered at (1, 2, 3) with radius 2 has which equation?
8. First octant
The first octant is characterized by which sign condition on the coordinates?
9. Completing the square for spheres
The equation x^2 + y^2 + z^2 - 2x - 4y - 6z = 2 describes a sphere. Completing the square gives center and radius what?
10. Third coordinate of sum
Adding (1, 2, 3) and (4, 5, 6) gives a vector whose z component equals what number? Answer with a number.
11. Box diagonal factor
The distance from the origin to (1, 1, 1) is root 3. As a number, what is under the root? Answer with a number.
12. Octant count
Three coordinate planes divide space into how many octants? Answer with a number.
13. Midpoint in space
The midpoint of the segment from (0, 0, 0) to (2, 4, 6) has which coordinates?
14. Right-handed orientation
The right-hand rule fixes the orientation by curling from x to y, the thumb pointing along which axis? Answer with one lowercase letter.
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