- Learn how to add and scale 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 in as dots in the line, plane, space, etc. We can also draw them as arrows. Since we have two geometric interpretations in mind, we now discuss the relationship between the two points of view.
Again, a point in is drawn as a dot.
A vector is a point in drawn as an arrow.
The difference is purely psychological: points and vectors are both just lists of numbers.
When we think of a point in 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.
Some authors use boldface letters to represent vectors, as in “”, or use arrows, as in “”. As it is usually clear from context if a letter represents a vector, we do not decorate vectors in this way.
Another way to think about a vector is as a difference between two points, or the arrow from one point to another. For instance, is the arrow from to
Here we learn how to add vectors together and how to multiply vectors by numbers, both algebraically and geometrically.
Addition and scalar multiplication work in the same way for vectors of length
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.
Geometrically, the difference of two vectors is obtained as follows: place the tail of and at the same point. Then is the vector from the head of to the head of For example,
Why? If you add to you get
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.
We can add and scale vectors in the same equation.
Let be scalars, and let be vectors in The vector in
is called a linear combination of the vectors with weights or coefficients
Geometrically, a linear combination is obtained by stretching / shrinking the vectors according to the coefficients, then adding them together using the parallelogram law.
A linear combination of a single vector is just a scalar multiple of So some examples include
The set of all linear combinations is the line through . (Unless in which case any scalar multiple of is again )
The set of all linear combinations of the vectors
is the line containing both vectors.
The difference between this and a previous example is that both vectors lie on the same line. Hence any scalar multiples of lie on that line, as does their sum.