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
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 .
The terminal point of this vector is and its initial point is . Therefore,
To express this in terms of the unit vectors,