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written by Tom Henderson
This interactive tutorial provides comprehensive help and practice in understanding vector quantities. It is organized into six sections: vector direction, vector addition, resultants, components, vector resolution, and component addition. It closes with relative velocity and riverboat problems, plus a discussion of the interdependence of perpendicular components of motion.  Each section explains a topic and provides exercises for learner self-assessment.  Images, animations, and graphs are placed throughout to illustrate the concepts.
Editor's Note: Fluency in vector operations requires practice, and this tutorial provides it.  Extra guidance is offered by clicking "Student Extras" at the top of the page.
Subjects Levels Resource Types
Classical Mechanics
- Motion in Two Dimensions
= Projectile Motion
Mathematical Tools
- Vector Algebra
- High School
- Middle School
- Lower Undergraduate
- Instructional Material
= Problem/Problem Set
= Tutorial
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- Physical Science
- Conceptual Physics
- Algebra-based Physics
- AP Physics
- Lesson Plan
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- New teachers
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© 1996 Tom Henderson
inclined plane, initial velocity, net force, relative velocity, resolution of vector components, resultants, vector components, vectors
Record Cloner:
Metadata instance created December 14, 2004 by Melanie Carter
Record Updated:
February 28, 2013 by Bruce Mason
Last Update
when Cataloged:
December 12, 2004
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AAAS Benchmark Alignments (2008 Version)

2. The Nature of Mathematics

2A. Patterns and Relationships
  • 9-12: 2A/H1. Mathematics is the study of quantities and shapes, the patterns and relationships between quantities or shapes, and operations on either quantities or shapes. Some of these relationships involve natural phenomena, while others deal with abstractions not tied to the physical world.

9. The Mathematical World

9B. Symbolic Relationships
  • 6-8: 9B/M2. Rates of change can be computed from differences in magnitudes and vice versa.
  • 9-12: 9B/H2b. Symbolic statements can be combined to look for values of variables that will satisfy all of them at the same time.
  • 9-12: 9B/H4. Tables, graphs, and symbols are alternative ways of representing data and relationships that can be translated from one to another.
  • 9-12: 9B/H5. When a relationship is represented in symbols, numbers can be substituted for all but one of the symbols and the possible value of the remaining symbol computed. Sometimes the relationship may be satisfied by one value, sometimes by more than one, and sometimes not at all.
9C. Shapes
  • 6-8: 9C/M9. Relationships exist among the angles between the sides of triangle and the lengths of those sides. For example, when two sides of a triangle are perpendicular, the sum of the squares of the lengths of those sides is equal to the square of the third side of the triangle.

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.

High School — Functions (9-12)

Trigonometric Functions (9-12)
  • F-TF.3 (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for ?/3, ?/4 and ?/6, and use the unit circle to express the values of sine, cosine, and tangent for ?—x, ?+x, and 2?—x in terms of their values for x, where x is any real number.

High School — Geometry (9-12)

Similarity, Right Triangles, and Trigonometry (9-12)
  • G-SRT.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

Common Core State Reading Standards for Literacy in Science and Technical Subjects 6—12

Key Ideas and Details (6-12)
  • RST.9-10.3 Follow precisely a complex multistep procedure when carrying out experiments, taking measurements, or performing technical tasks, attending to special cases or exceptions defined in the text.
Craft and Structure (6-12)
  • RST.11-12.4 Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 11—12 texts and topics.
Range of Reading and Level of Text Complexity (6-12)
  • RST.11-12.10 By the end of grade 12, read and comprehend science/technical texts in the grades 11—CCR text complexity band independently and proficiently.

This resource is part of a Physics Front Topical Unit.

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

As instructors, we may forget that certain representations (like vector arrows) seem like a foreign language to beginning students. This thoughtfully-crafted tutorial introduces vector diagrams in kid-friendly language and extends the learning to interactive practice problems with answers provided.

Link to Unit:
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AIP Format
T. Henderson, (1996), WWW Document, (https://www.physicsclassroom.com/Class/vectors/U3L1a.cfm).
T. Henderson, The Physics Classroom: Vectors - Fundamentals and Operations (1996), <https://www.physicsclassroom.com/Class/vectors/U3L1a.cfm>.
APA Format
Henderson, T. (2004, December 12). The Physics Classroom: Vectors - Fundamentals and Operations. Retrieved June 22, 2024, from https://www.physicsclassroom.com/Class/vectors/U3L1a.cfm
Chicago Format
Henderson, Tom. The Physics Classroom: Vectors - Fundamentals and Operations. December 12, 2004. https://www.physicsclassroom.com/Class/vectors/U3L1a.cfm (accessed 22 June 2024).
MLA Format
Henderson, Tom. The Physics Classroom: Vectors - Fundamentals and Operations. 1996. 12 Dec. 2004. 22 June 2024 <https://www.physicsclassroom.com/Class/vectors/U3L1a.cfm>.
BibTeX Export Format
@misc{ Author = "Tom Henderson", Title = {The Physics Classroom: Vectors - Fundamentals and Operations}, Volume = {2024}, Number = {22 June 2024}, Month = {December 12, 2004}, Year = {1996} }
Refer Export Format

%A Tom Henderson %T The Physics Classroom: Vectors - Fundamentals and Operations %D December 12, 2004 %U https://www.physicsclassroom.com/Class/vectors/U3L1a.cfm %O text/html

EndNote Export Format

%0 Electronic Source %A Henderson, Tom %D December 12, 2004 %T The Physics Classroom: Vectors - Fundamentals and Operations %V 2024 %N 22 June 2024 %8 December 12, 2004 %9 text/html %U https://www.physicsclassroom.com/Class/vectors/U3L1a.cfm

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The Physics Classroom: Vectors - Fundamentals and Operations:

Supplements PhET Simulation: Motion in 2D

A simple simulation for beginners designed to help them differentiate velocity and acceleration vectors. Appropriate for middle school and high school.

relation by Caroline Hall
Has Student Extra The Physics Classroom: Vector Direction

Need to see it? View the Vector Direction animation at the Multimedia Physics Studios.

relation by Tom Henderson
Has Teaching Guide Navigational Vectors

Explore nine lessons from a two-week unit on vectors by the Center for Innovation in  Science and Engineering Education (CIESE).

relation by Tom Henderson
Has Student Extra Vector Direction Help

Practice makes perfect. Try the Vector Direction page in the Physics Help section.

relation by Tom Henderson
Has Teaching Guide Curriculum Corner

Learning requires action. Give your students this sense-making activity from The Curriculum Corner.

relation by Tom Henderson

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