Vectors – Scalar Multiplication – Dot Product

Dot Product

The scalar dot product of the vectors u = (u₁, u₂, u₃) = u_i+ u_j+ u_k and v = (v₁, v₂, v₃) = v_i+ v_j+ v_k which is a scalar definition to be

u * v = u₁v₁ + u₂v₂ + u₃v₃

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.