Vector Operations

Sum of Vectors §

To add vectors, just add their corresponding components. Given and , the sum of the vectors is

Graphically, we can see that adding vectors means connecting the terminal point of one to the tail of the other.

Addition is commutative.

Difference of Vectors §

Given and , the difference of the vectors is

Subtraction is not commutative.

Multiplying by a scalar §

Multiplying a vector by a scalar is just like matrix scalar multiplication. For instance, if we multiply by 2, we get:

Graphically, we can see that the resulting column vector has the same direction, but its magnitutde is scaled by the absolute value of the scalar.

Multiplying a vector by a vector §

The product of two vectors is called dot product, and we find it by summing the products of the individual components.

We can also find the dot product when we write the vectors as matrices. So if we wrote as the matrix and as the matrix , then we could have written the dot product as

When we express the vectors as matrices and then multiply them, it’s important to multiply them as a row matrix first, multiplied by a column matrix second.

Otherwise we’d get a result that is completely different from the dot product of the two vectors.