Objectives
- 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.
Section2.1Vectors¶ permalink
Subsection2.1.1Vectors in
Recall that an element is called a point. Geometrically we may think of a point as a little dot floating in space. There is a useful alternative perspective given by thinking of an element of as an arrow extending from the origin to the point With this in mind we call a vector.
Points and Vectors
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 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 the velocity vector of a car as emanating from the car.
Note
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
Subsection2.1.2Vector Algebra and Geometry
Here we learn how to add vectors together and how to multiply vectors by numbers, both algebraically and geometrically.
Vector addition and scalar multiplication
There are two important operations one may perform with vectors:
- We can add two vectors together:
- We can scale a vector by a real number We call a scalar to distinguish it from a vector. If is a vector and is a scalar, then is called a scalar multiple of
Addition and scalar multiplication work in the same way for vectors of any length.
The Parallelogram Law for Vector Addition
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.
Vector Subtraction
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
Scalar Multiplication
A scalar multiple of a vector has the same (or opposite) direction, but its length is scaled. For instance, is the vector pointing in the same direction as but twice as long, and is the vector in the opposite direction of and half as long. Note that the set of all scalar multiples of a (nonzero) vector is a line.
Subsection2.1.3Linear Combinations
Combining addition and scalar multiplication gives us linear combinations.
Definition
Let be scalars, and let be vectors in The vector in defined by
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.
Example(Linear Combinations of a Single Vector)
A linear combination of a single vector is just a scalar multiple of So some examples include
The set of all linear combinations of just itself is the line through . (Unless in which case any scalar multiple of is again )
Example(Linear Combinations of Collinear Vectors)
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.