AP Precalculus Section 4.11 Example: Find the Area of a Triangle Using Vectors and Matrices

NOTE: Unit 4 is not included in the official AP Precalculus exam, but is still included in the curriculum.

Random AP Precalculus Problems (I found on the Internet). These are not official AP Collegeboard examples, but they will definitely get the job done!

To find the area of a triangle using vectors and matrices given the vertices as coordinates, you can follow these steps:

1. **Determine the Vectors**: Calculate the vectors representing two sides of the triangle using the coordinates of the vertices. For example, if you have vertices \(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\), you would calculate the vectors \(\vec{AB}\) and \(\vec{AC}\).

2. **Compute the Cross Product**: Take the cross product of the two vectors obtained in step 1. The magnitude of this cross product will be twice the area of the triangle.

3. **Calculate Area**: Divide the magnitude of the cross product by 2 to get the area of the triangle.

Mathematically, the formula to find the area of a triangle using vectors and matrices is given by:

\[
\text{Area} = \frac{1}{2} \| \vec{AB} \times \vec{AC} \|
\]

Where:
- \(\vec{AB}\) and \(\vec{AC}\) are the vectors formed by the coordinates of the vertices of the triangle.
- \(\times\) denotes the cross product.
- \(\| \vec{AB} \times \vec{AC} \|\) denotes the magnitude of the cross product.

The Topics covered in AP Precalculus are...

1.1 Change in Tandem
1.2 Rates of Change
1.3 Rates of Change in Linear and Quadratic Functions
1.4 Polynomial Functions and Rates of Change
1.5 Polynomial Functions and Complex Zeros
1.6 Polynomial Functions and End Behavior
1.7 Rational Functions and End Behavior
1.8 Rational Functions and Zeros
1.9 Rational Functions and Vertical Asymptotes
1.10 Rational Functions and Holes
1.11 Equivalent Representations of Polynomial and Rational Expressions
1.12 Transformations of Functions
1.13 Function Model Selection and Assumption Articulation
1.14 Function Model Construction and Application
2.1 Change in Arithmetic and Geometric Sequences
2.2 Change in Linear and Exponential Functions
2.3 Exponential Functions
2.4 Exponential Function Manipulation
2.5 Exponential Function Context and Data Modeling
2.6 Competing Function Model Validation
2.7 Composition of Functions
2.8 Inverse Functions
2.9 Logarithmic Expressions
2.10 Inverses of Exponential Functions
2.11 Logarithmic Functions
2.12 Logarithmic Function Manipulation
2.13 Exponential and Logarithmic Equations and Inequalities
2.14 Logarithmic Function Context and Data Modeling
2.15 Semi-log Plots
3.1 Periodic Phenomena
3.2 Sine, Cosine, and Tangent
3.3 Sine and Cosine Function Values
3.4 Sine and Cosine Function Graphs
3.5 Sinusoidal Functions
3.6 Sinusoidal Function Transformations
3.7 Sinusoidal Function Context and Data Modeling
3.8 The Tangent Function
3.9 Inverse Trigonometric Functions
3.10 Trigonometric Equations and Inequalities
3.11 The Secant, Cosecant, and Cotangent Functions
3.12 Equivalent Representations of Trigonometric Functions
3.13 Trigonometry and Polar Coordinates
3.14 Polar Function Graphs
3.15 Rates of Change in Polar Functions
4.1 Parametric Functions
4.2 Parametric Functions Modeling Planar Motion
4.3 Parametric Functions and Rates of Change
4.4 Parametrically Defined Circles and Lines
4.5 Implicitly Defined Functions
4.6 Conic Sections
4.7 Parametrization of Implicitly Defined Functions
4.8 Vectors
4.9 Vector-Valued Functions
4.10 Matrices
4.11 The Inverse and Determinant of a Matrix
4.12 Linear Transformations and Matrices
4.13 Matrices as Functions
4.14 Matrices Modeling Contexts

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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