# Explainer: Vectors in Terms of Fundamental Unit Vectors

In this explainer, we will learn how to write vectors in component form using fundamental unit vectors.

The two unit vectors are

They are “unit” vectors because they each have magnitude 1:

Using scalar multiplication and vector addition, we can express any vector in terms of and . Consider the vector

Using the definition of vector addition, we can express as a sum of a horizontal and vertical vector as follows

Then using the property of scalar multiplication, we can rewrite

Hence,

The general case is no harder. Suppose that the components of are and . Then,

Observe that by convention we convert the addition of a negative multiple of a vector to subtraction, so

### Example 1: Expressing the Components of a Vector in Terms of the Standard Unit Vectors

Express the vector using the unit vectors and .

If the vector to be expressed is geometric, we first write it in component form.

### Example 2: Expressing a Vector in Terms of the Standard Unit Vectors

The given figure shows a vector in a plane. Express this vector in terms of the unit vectors and .