published by
the Department of Physics, University of Guelph

This page offers a straightforward tutorial on the fundamentals of vector operations. It is an illustrated guide to vector subtraction/addition, vector resolution, and multiplication of two vectors. It is part of a larger collection of tutorials, intended for independent student use.

Editor's Note:While not the flashiest resource on the web, it is accurate, comprehensible, and could serve well as textbook supplementation or as free content support for science teachers. It can also be assigned as extra student help to be done outside class.

Common Core State Standards for Mathematics Alignments

High School — Number and Quantity (9-12)

Vector and Matrix Quantities (9-12)

N-VM.1 (+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).

N-VM.2 (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

N-VM.3 (+) Solve problems involving velocity and other quantities that can be represented by vectors.

N-VM.4.a Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

N-VM.4.b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

N-VM.4.c Understand vector subtraction v — w as v + (—w), where —w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.

N-VM.5.a Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v_{x}, v_{y}) = (cv_{x}, cv_{y}).

This resource is part of a Physics Front Topical Unit.

Topic: Kinematics: The Physics of Motion Unit Title: Vectors

This award-winning web tutorial is a great choice for the crossover teacher who wants a refresher on vectors and their properties. Included is an introduction to free-body diagrams, example problems, a series of self-paced questions, and related interactive simulations.

<a href="https://www.compadre.org/precollege/items/detail.cfm?ID=7754">Department of Physics, University of Guelph. Guelph Physics Tutorials: Vectors. Guelph: Department of Physics, University of Guelph, March 29, 2006.</a>

Guelph Physics Tutorials: Vectors (Department of Physics, University of Guelph, Guelph, 2006), <http://www3.physics.uoguelph.ca/tutorials/vectors/vectors.html>.

Guelph Physics Tutorials: Vectors. (2006, March 29). Retrieved May 28, 2023, from Department of Physics, University of Guelph: http://www3.physics.uoguelph.ca/tutorials/vectors/vectors.html

Department of Physics, University of Guelph. Guelph Physics Tutorials: Vectors. Guelph: Department of Physics, University of Guelph, March 29, 2006. http://www3.physics.uoguelph.ca/tutorials/vectors/vectors.html (accessed 28 May 2023).

Guelph Physics Tutorials: Vectors. Guelph: Department of Physics, University of Guelph, 2006. 29 Mar. 2006. 28 May 2023 <http://www3.physics.uoguelph.ca/tutorials/vectors/vectors.html>.

@misc{
Title = {Guelph Physics Tutorials: Vectors},
Publisher = {Department of Physics, University of Guelph},
Volume = {2023},
Number = {28 May 2023},
Month = {March 29, 2006},
Year = {2006}
}

%T Guelph Physics Tutorials: Vectors %D March 29, 2006 %I Department of Physics, University of Guelph %C Guelph %U http://www3.physics.uoguelph.ca/tutorials/vectors/vectors.html %O text/html

%0 Electronic Source %D March 29, 2006 %T Guelph Physics Tutorials: Vectors %I Department of Physics, University of Guelph %V 2023 %N 28 May 2023 %8 March 29, 2006 %9 text/html %U http://www3.physics.uoguelph.ca/tutorials/vectors/vectors.html

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