Understanding the Dot Product: Geometric & Algebraic Properties

1. Chapter 13 Section 13.3 The Dot Product

2. Dot Product The dot product of two vectors u and v is a number (a scalar) that can be computed in the following ways: Geometrically the dot product gives information about the angle  between the vectors u and v, specifically: u  v Orthogonal Vectors If u and v are nonzero vectors the only way that is to have which means that (or ). This means the vectors are perpendicular which we call orthogonal. The vector u is orthogonal to vector v if and only if . Algebraic Properties of Dot Product Let u, v, and w be vectors and r a scalar. and if and only if (Here  is the zero vector.) If u and v are nonzero vectors if and only if

3. Example Find the angle between the vectors u and v given as: and Projections The projection of a vector u onto a non zero vector v is a vector parallel to v whose difference with u is orthogonal to v. To derive a formula for this let h be the length that v must be rescaled to get an orthogonal vector. Multiply the unit vector in v’s direction by h to get the projection. u  v h Example

4. Example Find all values for c so that the vectors u and v given to the right are perpendicular. The idea is to find the dot product of u and v and set it equal to zero. Setting equal to zero and solving: We get the solutions and Example Show that for any two nonzero vectors u and v the two vectors v and are perpendicular vectors. The formula for Now take the dot product of v and and simplify it.