Photo above: Alan Saller - A water-filled ornament shows a bullet tearing through the glass.

## Topic 4 - Vectors (16 hours)

The aim of this topic is to provide an elementary introduction to vectors, including both algebraic and geometric approaches. The use of dynamic geometry software is extremely helpful to visualize situations in three dimensions.

## Vectors - What You Should Know

The document "vectors_-what_you_should_know.pfd" below is where you will find key descriptions of the content of Vectors. The document outlines the mathematical concepts that make up the content of the topic. Some examples are provided that illustrate how these concepts have been presented in past exams, along with some commentary towards a solution to the example. There is also some "Be prepared" comments that focus your attention on common issues to help you prepare more thoroughly.

vectors_-_what_you_should_know.pdf | |

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3d_vectors_example_problems.pdf | |

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## Dancing Honeybee Using Vector Calculus to Communicate.

Vectors are used extensively in a branch of Physics called Mechanics. They are used to represent quantities such as displacement, force, weight, velocity and momentum. In Mathematics we are primarily interested in vectors as representation of displacements and velocities.

## Vector Introduction PowerPoint

vectors_intro_2x2_powerpoint.ppt | |

File Size: | 1569 kb |

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## Vectors

vectors_worksheet_a.pdf | |

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vectors_worksheet_a_soln.pdf | |

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## Cartesian Components of a Vector

cartesian_components_of_a_vector_worksheet_b.pdf | |

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cartesian_components_of_a_vector_worksheet_b_soln.pdf | |

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## The Dot Product

Let's say we have two vectors, A and B, as shown to the left. The values |A| and |B| represent the lengths of vectors A and B, respectively, and

A · B = |A| * |B| * cos(

That is to say, the dot product of two vectors will be equal to the cosine of the angle between the vectors, times the lengths of each of the vectors.

Given the characteristics of the cosine function, we can deduce three possible conditions:

*Θ*is the angle between the two vectors. The dot product of vectors A and B will have the following relationship to these values:A · B = |A| * |B| * cos(

*Θ*)That is to say, the dot product of two vectors will be equal to the cosine of the angle between the vectors, times the lengths of each of the vectors.

*Angular Domain of Dot Product:*Given the characteristics of the cosine function, we can deduce three possible conditions:

- If A and B are perpendicular (at 90 degrees to each other), the result of the dot product will be zero, because cos(
*Θ*) will be zero. - If the angle between A and B are less than 90 degrees, the dot product will be positive (greater than zero), as cos(
*Θ*) will be positive, and the vector lengths are always positive values. - If the angle between A and B are greater than 90 degrees, the dot product will be negative (less than zero), as cos(
*Θ*) will be negative, and the vector lengths are always positive values.

## Getting an Angle on Vectors Investigation

getting_an_angle_on_vectors.pdf | |

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## Angles Between Two Vectors PowerPoint

vectors_2d_angles_powerpoint.ppt | |

File Size: | 1715 kb |

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## Perpendicular and Parallel Vectors PowerPoint

vectors_parallel_intersecting_skew_powerpoint.ppt | |

File Size: | 1462 kb |

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## The Scalar Product

the_scalar_product_worksheet_d.pdf | |

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the_scalar_product_worksheet_d_soln.pdf | |

File Size: | 141 kb |

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## Vectors in Three Dimensions PowerPoint

vectors_3d_powerpoint.ppt | |

File Size: | 1454 kb |

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sl_maths_vectors_-_revision.ppt | |

File Size: | 2792 kb |

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## Vector Equation of a Straight Line PowerPoint

vectors_equation_of_a_line_powerpoint.ppt | |

File Size: | 1643 kb |

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## Vector Equation of a Straight Line

vector_equation_of_a_straight_line_worksheet_c.pdf | |

File Size: | 115 kb |

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vector_equation_of_a_straight_line_worksheet_c_soln.pdf | |

File Size: | 128 kb |

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## Vector Lines in Space PowerPoint

vectors_lines_in_space_powerpoint.ppt | |

File Size: | 1308 kb |

File Type: | ppt |

## Mixed Exam-Style Questions on Vectors

mixed_exam-style_questions_on_vectors_worksheet_e.pdf | |

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mixed_exam-style_questions_on_vectors_worksheet_e_soln.pdf | |

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mixed_exam-style_questions_on_vectors_worksheet_f.pdf | |

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mixed_exam-style_questions_on_vectors_worksheet_f_soln.pdf | |

File Size: | 105 kb |

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