## Dot Product

The scalar dot product of the vectors u = (u_{1}, u_{2}, u_{3}) = u_{i}+ u_{j}+ u_{k} and v = (v_{1}, v_{2}, v_{3}) = v_{i}+ v_{j}+ v_{k} which is a scalar definition to be

u * v = u_{1}v_{1} + u_{2}v_{2} + u_{3}v_{3}

To find the angle between vectors, the following formula is used:

Rearranging the equation to solve for 0,

**Example 1 – Find the dot product of u = <1, 2, 3> and v = <4, 5, 6>**

u * v = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 22

**Example 2 – Find the dot product of a = 5i + 6j + 7k and b = 8i + 9j -10k.**

a * b = 5*8 + 6*9 + 7*(-10) = 40 + 54 – 70 = 24

**Example 3 – Find the angle between the vectors from the previous example.**

1. Find the dot product a * b

From the previous problem, the dot product a * b = 24.

2. Find |a| and |b|.

3. Plug values into the dot product equation:

The angle between a and b is 81.544 degrees.