Unit Vectors

Unit Vectors

By Elliot Moutra

Unit Vectors

A vector that has a magnitude of 1 unit is called a unit vector. it is sometimes useful to describe a nonzero vector v as a scalar multiple of a unit vector u.

​|v| => must be computed as a simplified radical

​If the component form = <v₁, v₂>

​Ex: Find a unit vector u with the same direction as v = <-2, 3>

​Qz: <-2[13]/13, 3[13]/13>

​Remember: v₁ = -2, v₂ = 3

​Ex: Find a unit vector u with the same direction as w = <6, -2>

​Qz: <3[10]/10, -[10]/10>

​Remember: v₁ = 6, v₂ = -2

​Vectors as a Linear Combination

If the component form of a vector is <a, b>, the linear combination would be written: ai + bj

• Ex: <2, 3> = 2i + 3j

• Ex: <4, -5> = 4i - 5j

• Ex: <-7, 7> = -7i + 7j

​Step 1: Find component form of AB

• <2 - 5, -7 - 5> = <-3, -12>

Step 2: Convert component form to linear combination

• <-3, -12> = -3i - 12j

​Qz: -3i - 12j