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1.5 MATRICES AND VECTORS

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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,

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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.

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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,

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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,

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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),

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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,