A point represents a position in a plane (2D) or space (3D).
Given an origin, axes, and scale, we can assign coordinates to each point.
Changing the origin translates the coordinate system. The same points now have new coordinates.
Changing the axes rotates the coordinate system, once again changing the coordinates of the points.
A vector is an object with a direction and a magnitude (length), but not a location.
These represent the same vector
These also represent the same vector
These vectors have different directions
These vectors also have different directions
These vectors have different lengths
Analogous to points, given axes and a scale, we can assign components to vectors.
Changing the origin (translation) does not affect the components of a vector.
Rotating the axes does change vector components.
By anchoring the vectors at the origin, we can associate vectors with points. The coordinates of the end-points equal the components of the corresponding vectors.
As long as we keep the origin fixed, we can often ignore the distinction between points and vectors.
Given the coordinates of a vector, we can use the Pythagorean Theorem to compute its length:
|v|2 = vx2 + vy2.
|v| = sqrt( vx2 + vy2 ).
For example, if v = [ 4, 3 ], then
|v| = sqrt( 4×4 + 3×3 ) = sqrt( 16 + 9 ) = sqrt( 25 ) = 5.
Multiplying a vector by a scalar multiplies the length of the vector. If the scalar is negative, the direction is reversed.
The sum of v and w is formed by placing the beginning of w at the end of v and drawing a new vector from the beginning of v to the end of w.