A vector is an equivalence class of directed line segments that share the same length and direction.
The coordinates of a vector may be computed from any representative directed line segment by subtracting the coordinates of its tail point from the coordinates of its head point.
The length of a vector is the square root of the sum of the squares of its coordinates.
If v and w are non-zero vectors, then v and w are parallel if and only if each is a scalar multiple of the other.
Two vectors v and w are perpendicular if their scalar product v . w is the number 0.
Let f(x) = A x + a where x is a column with 3 coordinates, a is the column with coordinates (3, -2, -1), and A is the 3 \times 3 matrix
| 1 | 2 | 1 |
| 2 | -2 | 8 |
| -2 | -1 | -5 |
Note that although f does not satisfy the conditions of the discussion of February 5, it is still sensible to speak of its image and of its fiber over a point of 3-space, using the definitions in that discussion.
What equation must be satisfied by a point in the image of f ?
What is the fiber of f over the point (4, 0, -3) ?
What is the fiber of f over the point (1, 8, -5) ?
Let a, b, c be the points with a as above, b = (4, 0, -3), and c = (5, -4, -2). Let
Show that
Show that the plane in 3-space containing a, b, and c is the plane that is parameterized by phi.
Find an equation for this plane.
How does the triangle with vertices a, b, c compare with the triangle in the (s, t)-plane having vertices (0, 0), (1,0), and (0,1) ?