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 .

### Answer

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 .

### Answer

The terminal point of this vector is and its initial point is . Therefore,

To express this in terms of the unit vectors,