## 1.5 MATRICES AND VECTORS

### 1.5.1 Vectors

• Vectors are often drawn with arrows, as shown below, • Cartesian notation is also a common form of usage. • Vectors can be added and subtracted, numerically and graphically, ### 1.5.2 Dot (Scalar) Product

• We can use a dot product to find the angle between two vectors • We can use a dot product to project one vector onto another vector. • We can consider the basic properties of the dot product and units vectors.   ### 1.5.3 Cross Product

• First, consider an example, • The basic properties of the cross product are, • When using a left/right handed coordinate system, • The properties of the cross products are, ### 1.5.4 Triple Product ### 1.5.5 Matrices

• Matrices allow simple equations that drive a large number of repetitive calculations - as a result they are found in many computer applications.

• A matrix has the form seen below, • Matrix operations are available for many of the basic algebraic expressions, examples are given below. There are also many restrictions - many of these are indicated.       • The eigenvalue of a matrix is found using, ### 1.5.6 Solving Linear Equations with Matrices

• We can solve systems of equations using the inverse matrix, • We can solve systems of equations using Cramer’s rule (with determinants), ### 1.5.7 Practice Problems

1. Perform the matrix operations below. 2. Perform the vector operations below, 4. Solve the following equations using any technique, 