# Engineering Science 1022A/B/Y Lecture Notes - Cartesian Coordinate System, Parallelogram Law, Unit Vector

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Published on 19 Sep 2012

Department

Engineering Science

Course

Engineering Science 1022A/B/Y

Professor

ES 1022y Engineering Statics Force Vectors

6

Force Vectors

In this course forces (acting along the vector axis), moments (rotation about the vector

axis), and position vectors (moving along the vector axis) are all vector quantities. A

force, moment or position vector has:

A vector quantity can be represented graphically by an arrow that shows its magnitude,

direction and sense.

Magnitude: characterized by size in some units, e.g. 34 N; represented by length of the

arrow according to some scale, say, 1 cm = 10 N ā 3.4 cm = 34 N.

Direction: the angle between a reference axis and the arrow's line of action.

Sense: indicated by the arrowhead (one of two possible directions)

A Word on Vector Notation

In the lecture notes and text book a vector quantity is indicated by a letter in boldface

type (F), while the magnitude of a vector is denoted by an italicized letter (F). For

handwritten work a vector is usually indicated by drawing an arrow above the letter

representing the vector, thus

Similarly, unit vectors can be denoted in handwritten work by drawing a hat symbol

above the letter to give

A magnitude, a direction, a sense, a point of

application

ES 1022y Engineering Statics Force Vectors

7

Vector Addition

Consider two force vectors A and B. We want to add them together to find the vector

sum, or resultant force vector, R such that

We can do this using one of two methods.

Parallelogram law:

If the two forces A and B are represented by the adjacent sides of a parallelogram, then

the diagonal of the parallelogram is equal to the vector sum of the two forces.

Triangle of forces:

Special case of the parallelogram law.

R = A + B = B +

A

ES 1022y Engineering Statics Force Vectors

8

As a special case, if the two vectors A and B are collinear, that is

the parallelogram law reduces to an algebraic or scalar addition

Where we have three or more forces we can either use repeated applications of the

parallelogram law or a force polygon to find the resultant force.

We can also use trigonometry to add two force vectors together using the sine and cosine

laws. Consider a triangle with sides of length A, B, and C, and corresponding interior

angles a, b, and c.

Sine law:

Cosine law:

Both vectors have the same line of action

## Document Summary

In this course forces (acting along the vector axis), moments (rotation about the vector axis), and position vectors (moving along the vector axis) are all vector quantities. A vector quantity can be represented graphically by an arrow that shows its magnitude, direction and sense. Magnitude: characterized by size in some units, e. g. 34 n; represented by length of the arrow according to some scale, say, 1 cm = 10 n 3. 4 cm = 34 n. Direction: the angle between a reference axis and the arrow"s line of action. Sense: indicated by the arrowhead (one of two possible directions) In the lecture notes and text book a vector quantity is indicated by a letter in boldface type (f), while the magnitude of a vector is denoted by an italicized letter (f). For handwritten work a vector is usually indicated by drawing an arrow above the letter representing the vector, thus.