Scalar product of vectors

If we have two vectors, and , we write the scalar product as:

Because of the big dot operator which we use to mean ‘scalar product’, it is also called the dot product.

Scalar product rule for 2D

For vectors and , then .

Scalar product rule for 3D

For 3D, it’s exactly the same.

For vectors and , then .

Geometric definition

Proof

• The origin is at point .
• Point is at .
• Point is at .
• let
• let
• let be the angle between and .
• The cosine rule is .

Applying the rules: finding the angle between two vectors

Find the angle between and

• • so
• • so
• So the angle between the two vectors is - we’ve solved it!!

Find the angle between and

• So the angle between the two vectors is :)

Find the angle between and

Checking if vectors are perpendicular

• If , then .
• So if the scalar product of two vectors is zero, then the vectors are perpendicular:

Key take-away: if , then .