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written by Carl Rod Nave
This page is an interactive tutorial that allows the user to choose among topics related to vector operations. The subjects are organized in flow charts that make it easy to move from one topic to a related item.  Vector resolution, addition, and product are covered in-depth, and the site gives conceptual support in practical applications such as torque, work, and magnetic force.
Subjects Levels Resource Types
Classical Mechanics
- Motion in Two Dimensions
Mathematical Tools
- Vector Algebra
- Lower Undergraduate
- High School
- Instructional Material
= Tutorial
- Reference Material
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- Learners
- Educators
- text/html
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© 2000 C. R. Nave
vector addition, vector product, vector resolution, vector sum, vectors
Record Cloner:
Metadata instance created October 24, 2006 by Caroline Hall
Record Updated:
January 10, 2014 by Caroline Hall
Last Update
when Cataloged:
September 10, 2006
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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.
2B. Mathematics, Science, and Technology
  • 9-12: 2B/H3. Mathematics provides a precise language to describe objects and events and the relationships among them. In addition, mathematics provides tools for solving problems, analyzing data, and making logical arguments.

4. The Physical Setting

4F. Motion
  • 6-8: 4F/M3a. An unbalanced force acting on an object changes its speed or direction of motion, or both.

9. The Mathematical World

9B. Symbolic Relationships
  • 9-12: 9B/H4. Tables, graphs, and symbols are alternative ways of representing data and relationships that can be translated from one to another.

Next Generation Science Standards

Disciplinary Core Ideas (K-12)

Forces and Motion (PS2.A)
  • The motion of an object is determined by the sum of the forces acting on it; if the total force on the object is not zero, its motion will change. The greater the mass of the object, the greater the force needed to achieve the same change in motion. For any given object, a larger force causes a larger change in motion. (6-8)
  • All positions of objects and the directions of forces and motions must be described in an arbitrarily chosen reference frame and arbitrarily chosen units of size. In order to share information with other people, these choices must also be shared. (6-8)

NGSS Science and Engineering Practices (K-12)

Using Mathematics and Computational Thinking (5-12)
  • Mathematical and computational thinking at the 9–12 level builds on K–8 and progresses to using algebraic thinking and analysis, a range of linear and nonlinear functions including trigonometric functions, exponentials and logarithms, and computational tools for statistical analysis to analyze, represent, and model data. Simple computational simulations are created and used based on mathematical models of basic assumptions. (9-12)
    • Use mathematical representations of phenomena to describe explanations. (9-12)

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.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(vx, vy) = (cvx, cvy).

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

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.
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AIP Format
C. Nave, (2000), WWW Document, (
C. Nave, Hyperphysics: Mechanics: Basic Vector Operations, (2000), <>.
APA Format
Nave, C. (2006, September 10). Hyperphysics: Mechanics: Basic Vector Operations. Retrieved February 28, 2017, from
Chicago Format
Nave, Carl Rod. Hyperphysics: Mechanics: Basic Vector Operations. September 10, 2006. (accessed 28 February 2017).
MLA Format
Nave, Carl Rod. Hyperphysics: Mechanics: Basic Vector Operations. 2000. 10 Sep. 2006. 28 Feb. 2017 <>.
BibTeX Export Format
@misc{ Author = "Carl Rod Nave", Title = {Hyperphysics: Mechanics: Basic Vector Operations}, Volume = {2017}, Number = {28 February 2017}, Month = {September 10, 2006}, Year = {2000} }
Refer Export Format

%A Carl Rod Nave
%T Hyperphysics: Mechanics: Basic Vector Operations
%D September 10, 2006
%O text/html

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%0 Electronic Source
%A Nave, Carl Rod
%D September 10, 2006
%T Hyperphysics: Mechanics: Basic Vector Operations
%V 2017
%N 28 February 2017
%8 September 10, 2006
%9 text/html

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