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 ij=1,0=0,1.and

They are “unit” vectors because they each have magnitude 1: ||=1+0=1,||=0+1=1.ij

Using scalar multiplication and vector addition, we can express any vector in terms of i and j. Consider the vector v=2,𝜋.

Using the definition of vector addition, we can express v as a sum of a horizontal and vertical vector as follows v=2,𝜋=2,0+0,𝜋.

Then using the property of scalar multiplication, we can rewrite v=21,0+𝜋0,1.

Hence, vij=2+𝜋.

The general case is no harder. Suppose that the components of v are 𝑎 and 𝑏. Then, vij=𝑎,𝑏=𝑎,0+0,𝑏=𝑎1,0+𝑏0,1=𝑎+𝑏.

Observe that by convention we convert the addition of a negative multiple of a vector to subtraction, so (2)+(3)=23.ijij

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

Express the vector 52,19 using the unit vectors i and j.

Answer

52,19=52,0+0,19=521,0+(19)0,1=5219.ij

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 i and j.

Answer

The terminal point of this vector is (4,8) and its initial point is (2,2). Therefore, 42,8(2)=2,10.

To express this in terms of the unit vectors, 2,10=2,0+0,10=(2)1,0+(10)0,1=2+10.ij

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