Vectors – Scalar Multiplication – Dot Product

Dot Product

The scalar dot product of the vectors u = (u1, u2, u3) = ui+ uj+ uk and v = (v1, v2, v3) = vi+ vj+ vk which is a scalar definition to be

u * v = u1v1 + u2v2 + u3v3

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

vectors 3

Rearranging the equation to solve for 0,

scalar

multiplication 1

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|.

dot product

3. Plug values into the dot product equation: scalar multiplication

angle

The angle between a and b is 81.544 degrees.