Learn how to add and scalar-multiply vectors in both algebraically and geometrically.
Understand linear combinations geometrically.
Pictures: vector addition, vector subtraction, linear combinations.
Vocabulary words:vector, linear combination.
We have been drawing points of as dots in the line, plane, space, etc. We can also draw them as arrows. Since we have two geometric interpretations in mind, in what follows we will call an ordered list of real numbers an element of
Points and Vectors
A point is an element of drawn as a point (a dot).
A vector is an element of drawn as an arrow.
The difference is purely psychological: points and vectors are both just lists of numbers.
When we think of an element of as a vector, we will usually write it vertically, like a matrix with one column:
We will also write for the zero vector.
Why make the distinction between points and vectors? A vector need not start at the origin: it can be located anywhere! In other words, an arrow is determined by its length and its direction, not by its location. For instance, these arrows all represent the vector
Unless otherwise specified, we will assume that all vectors start at the origin.
Vectors makes sense in the real world: many physical quantities, such as velocity, are represented as vectors. But it makes more sense to think of the velocity of a car as being located at the car.
Geometrically, the sum of two vectors is obtained as follows: place the tail of at the head of Then is the vector whose tail is the tail of and whose head is the head of Doing this both ways creates a parallelogram. For example,
Why? The width of is the sum of the widths, and likewise with the heights.
A scalar multiple of a vector has the same (or opposite) direction, but a different length. For instance, is the vector in the direction of but twice as long, and is the vector in the opposite direction of but half as long. Note that the set of all scalar multiples of a (nonzero) vector is a line.