Basic Vector Operations

Now that we have a good understanding of what vectors are, we can start talking about some basic operations we can perform on vectors. These operations, in addition to being quite intuitive, are also the building blocks for defining vector spaces and more advanced concepts in linear algebra.

Vector Addition

The first operation we will discuss is vector addition. Given two vectors and , we can add them together to get a new vector .

Addition: Geometric Interpretation

To add two vectors, we can place the tail of the second vector at the tip of the first vector. Then, draw a new vector from the tail of the first vector to the tip of the second vector. This is best seen visually. Consider the two vectors and :

Place the tail of at the tip of , and draw a new vector from the tail of to the tip of :

This should make sense if you, again, think of vectors as instructions to move from one point to another. This is also very similar to how we would add two normal numbers together. For instance, if we have the numbers and , we would move 3 units to the right and then 2 units to the left, ending up at .

Addition: Algebraic Interpretation

To add two vectors together, we can simply add their components together. For example, if we have the vectors and , we can add them together to get:

This should make sense with the instructions analogy; going 2 units right and 1 unit up, and then going 1 unit left and 2 units up, will result in going 1 unit right and 3 units up. We can actually see this visually as well by decomposing each vector into its components:

Properties of Vector Addition

Vector addition has a few properties that are worth noting.

1. Commutativity: The order in which we add vectors does not matter. If we think geometrically, this is because the order in which we move does not matter.

2. Associativity: The way we group vectors when adding them does not matter. This is because we can think of adding vectors as moving from one point to another, and the order in which we move does not matter.

Scalar Multiplication

The second operation we will discuss is scalar multiplication.

Given a vector and a scalar , we can multiply the vector by the scalar to get a new vector .

Multiplication: Geometric Interpretation

To multiply a vector by a scalar, we can simply stretch or shrink the vector by the scalar. This has the effect of changing the magnitude of the vector, but not the direction.

Let's scale a vector by a factor of . This has the effect of doubling its length:

Multiplying a vector by a negative scalar has the effect of reversing the direction of the vector. Here's a playground where you can experiment with scalar multiplication:

Scalar Multiplication Playground

Options
Scale

Multiplication: Algebraic Interpretation

To multiply a vector by a scalar, we can simply multiply each component of the vector by the scalar.

For example, if we have the vector and we multiply it by , we get:

This should make sense with the instructions analogy; going 2 units right and 1 unit up, and then doubling that, will result in each direction being doubled.

Properties of Scalar Multiplication

Scalar multiplication also has a few properties that are worth noting:

1. Distributivity over Vector Addition: Multiplying a vector by a scalar and then adding another vector is the same as adding the vectors first and then multiplying by the scalar.

2. Distributivity over Scalar Addition: Multiplying a vector by the sum of two scalars is the same as multiplying the vector by each scalar separately and then adding the results.

3. Associativity with Scalar Multiplication: Multiplying a vector by the product of two scalars is the same as multiplying the vector by each scalar separately and then multiplying the results.

A good exercise would be to show why each property makes sense geometrically.

Unit Vectors

Sometimes, we use a vector to describe just the direction of a vector, without the magnitude. It's very useful to have a vector with a certain direction we want to describe, but with a magnitude of 1. These vectors are called unit vectors.

A unit vector in the direction of a vector is denoted by .

These are very useful - for instance, let's say we want to describe a vector with a length of but with the same direction as a unit vector . We can simply multiply by to get .

To make a unit vector from another vector, we can simply divide the vector by its magnitude:

Doing this has the result of scaling a vector down by whatever its magnitude is, resulting in a vector with a magnitude of . We will see the importance of unit vectors in the few pages.

Summary and Next Steps

In this page, we discussed two basic operations we can perform on vectors: addition and scalar multiplication. These operations are quite intuitive and are the building blocks for defining more advanced concepts in linear algebra.

Here are the key points to remember:

• Vector addition involves adding two vectors together to get a new vector:

  • Geometrically, this is like placing the tail of the second vector at the tip of the first vector and drawing a new vector from the tail of the first vector to the tip of the second vector.
  • Algebraically, this is like adding the components of the vectors together.

• Scalar multiplication involves multiplying a vector by a scalar to get a new vector:

  • Geometrically, this is like stretching or shrinking the vector by the scalar.
  • Algebraically, this is like multiplying each component of the vector by the scalar.

• Unit vectors are vectors with a magnitude of that describe a certain direction. They can be made from other vectors by dividing the vector by its magnitude:

With the most fundamental operations out of the way, we can now start discussing some more advanced concepts surrounding vectors. In the next page, we will discuss linear combinations, span, subspaces, the basis vectors, and linear independence. A lot of these concepts will build on what we have discussed here, so make sure you have a good understanding of these basic operations before moving on.