Dot Product
The dot product or scalar product of two vectors is the product of their magnitudes multiplied by the cosine of the angle that they form.
It can also be expressed as:
Example
Find the dot product of two vectors whose coordinates in an orthonormal basis are: (1, 1/2, 3) and (4, −4, 1).
(1, 1/2, 3) · (4, −4, 1) = 1 · 4 + (1/2) · (−4) + 3 · 1 = 4 −2 + 3 = 5
Magnitude of a Vector
Example
Find the magnitude of a vector with coordinates 
= (−3, 2, 5) in an orthonormal basis.
Angle between Two Vectors
Example
Determine the angle between the vectors = (1, 2, −3) and 
= (−2, 4, 1).
Orthogonal Vectors
Two vectors are orthogonal or perpendicular if their dot product is zero.
Example
Calculate the x and y values for the vector (x, y, 1) that is orthogonal to the vectors (3, 2, 0) and (2, 1, −1).
Properties of the Dot Product
1Commutative
2 Associative
3 Distributive
4
Geometric Interpretation of the Dot Product
Scalar Projection
Vector Projection
The vector projection is the unit vector of by the scalar projection of u on v.
The scalar projection of u on v is the magnitude of the vector projection of u on v.
Exercise
Given the vectors 
and 
, calculate:
1. The magnitudes of and ·
2. The dot product of and ·
3. The angle.
4. The value of m for which vectors and 
are orthogonal.
