Chapter 12 – Vectors and the Geometry of Space 12.3 – The Dot Product 12.3 – The Dot Product

Definition – Dot Product Note: The result is not a vector. It is a real number, a scalar. Sometimes the dot product is called the scalar product or inner product. 12.3 – The Dot Product

Example 1 – pg.806 # 8 Find a  b a = 3i + 2j - k b = 4i + 5k 12.3 – The Dot Product

Properties of the Dot Product

Theorem – Dot Product The dot product can be given a geometric interpretation in terms of the angle  between a and b. 12.3 – The Dot Product

Applying Law of Cosines We can apply the Law of Cosines to the triangle OAB and get the following formulas: 12.3 – The Dot Product

Corollary – Dot Product 12.3 – The Dot Product

Example 2 – pg. 806 # 18 Find the angle between the vectors. (First find an exact expression then approximate to the nearest degree.) a = <4, 0, 2> b = <2, -1, 0> 12.3 – The Dot Product

Orthogonal Vectors Two nonzero a and b are called perpendicular or orthogonal if the angles between them is  = /2. 12.3 – The Dot Product

Hints The dot product is a way of measuring the extent to which the vectors point in the same direction. If the dot product is positive, then the vectors point in the same direction. If the dot product is 0, the vectors are perpendicular. If the dot product is negative, the vectors point in opposite directions. 12.3 – The Dot Product

Visualization The Dot Product of Two Vectors 12.3 – The Dot Product

Example 3 For what values of b are the given vectors orthogonal? <-6, b, 2> <b, b2, b> 12.3 – The Dot Product

Definition – Directional Angles The directional angles of a nonzero vector a are the angles , , and  in the interval from 0 to pi that a makes with the positive axes. 12.3 – The Dot Product

Definition – Direction Cosines We get the direction cosines of a vector a by taking the cosines of the direction angles. We get the following formulas 12.3 – The Dot Product

Continued 12.3 – The Dot Product

Example 4 pg. 806 #35 Find the direction cosines and direction angles of the vector. Give the direction angles correct to the nearest degree. i – 2j – 3k 12.3 – The Dot Product

Definition - Vector Projection If S is the foot of the perpendicular from R to the line containing , then the vector with representation is called the vector projection of b onto a and is denoted by projab. (think of it as a shadow of b.) 12.3 – The Dot Product

Definition continued 12.3 – The Dot Product

Visualization Vector Projections 12.3 – The Dot Product

Definition – Scalar Projection The scalar projection or component of b onto a is defined to be the signed magnitude of the vector projection, which is the number |b|cos, where  is the angle between a and b. This is denoted by compab. 12.3 – The Dot Product

Definition continued 12.3 – The Dot Product

Example 5 – pg807 #42 Find the scalar and vector projections of b onto a. a = <-2, 3, -6> b = <5, -1, 4> 12.3 – The Dot Product

More Examples The video examples below are from section 12.3 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. Example 1 Example 3 Example 6 12.3 – The Dot Product

Demonstrations Feel free to explore these demonstrations below. The Dot Product Vectors in 3D Vector Projections 12.3 – The Dot Product