Lesson Explainer: Vectors in Terms of Fundamental Unit Vectors Mathematics

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

The two unit vectors are ⃑𝑖=(1,0)⃑𝑗=(0,1).and

They are β€œunit” vectors because they each have magnitude 1: ‖‖⃑𝑖‖‖=√1+0=1,‖‖⃑𝑗‖‖=√0+1=1.

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

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

Then using the property of scalar multiplication, we can rewrite ⃑𝑣=2(1,0)+πœ‹(0,1).

Hence, ⃑𝑣=2⃑𝑖+πœ‹βƒ‘π‘—.

The general case is no harder. Suppose that the components of ⃑𝑣 are π‘Ž and 𝑏. Then, ⃑𝑣=(π‘Ž,𝑏)=(π‘Ž,0)+(0,𝑏)=π‘Ž(1,0)+𝑏(0,1)=π‘Žβƒ‘π‘–+𝑏⃑𝑗.

Observe that by convention we convert the addition of a negative multiple of a vector to subtraction, so (βˆ’2)⃑𝑖+(βˆ’3)⃑𝑗=βˆ’2βƒ‘π‘–βˆ’3⃑𝑗.

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

Express the vector ο€Όβˆ’52,βˆ’19 using the unit vectors ⃑𝑖 and ⃑𝑗.

Answer

ο€Όβˆ’52,βˆ’19=ο€Όβˆ’52,0+(0,βˆ’19)=ο€Όβˆ’52(1,0)+(βˆ’19)(0,1)=βˆ’52βƒ‘π‘–βˆ’19⃑𝑗.

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,βˆ’2). Therefore, (4βˆ’2,8βˆ’(βˆ’2))=(2,10).

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

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