Addition of Coplanar Forces - Using Cartesian Vector Notation

Useful book to have

Schaum's Outline of Engineering Mechanics: Statics

For problems and solutions on addition of coplanar forces click HERE.

Given the coplanar forces F₁, F₂ and F₃ below, find the resultant force F_R = F₁ + F₂ + F₃

Step 1:

Represent each force as a Cartesian vector:

Step 2:

Finally, determine the resultant force: 

Addition of Coplanar Forces – Using Scalar Notation

For problems and solutions on addition of coplanar forces click HERE.

Given the coplanar forces F₁, F₂ and F₃ below, find the resultant force F_R = F₁ + F₂ + F₃

Step 1:

Resolve the forces into their rectangular components i.e.:





Step 2:

Determine the resultant force in the x-axis and y-axis independently i.e.:



Step 3:

Finally determine the magnitude of the resultant force F_R and its direction θ :

For problems and solutions on addition of coplanar forces click HERE.

Coplanar Forces and their Rectangular Components

Coplanar forces: a set of forces is coplanar if they all lie in the same geometric plane.  In the diagram below forces F₁, F₂ and F₃ are coplanar because the all lie in the x-y plane.  Refer to the image below.

Coplanar forces can be resolved into components along the x and y axes.  These component forces are called rectangular components.  Rectangular components can be represented in two ways:

1. Scalar notation


2. Cartesian vector notation

Scalar Notation: When using the scalar notation the rectangular components of force F is written F_x and F_y.

• F_x represents the component in the x-axis


• F_y represents the component in the y-axis

Refer to the image below.

Cartesian Vector Notation: When using the Cartesian vector notation, the rectangular components are represented in terms of unit vectors i and j.  Unit vectors are vectors that have a magnitude of one and they are used to represent direction.  In this case:

• i represents direction in x-axis


• j represents direction in y-axis

Refer to the image below.

Vector Subtraction and Resolution of a Vector

Vector Subtraction

Given vectors A and B as follows:



The parallelogram for vector subtraction takes the following form:





Resolution of a Vector

A vector may be resolved into to components having known lines of action using the parallelogram law:

Given the vector R below:



Extend parallel lines from the head of vector R to form components in a and b axis



The resolution of vector R into vector A and vector B is as follows:

Two Dimenstional Vector Addition - The Parallelogram Law

For problems and solutions on the parallelogram law, click HERE.

Two vectors of the same type (i.e. two force vectors or two velocity vectors) can be added together  to obtain a single resultant vector.

Addition of two vectors can be done using the parallelogram law.

Given two vectors A and B find the resultant vector R = A + B



Step 1:

Join vectors A and B at their tails:



Step 2:

Draw parallel lines from the head of each vector to form a parallelogram:



Step 3:

To obtain the resultant of the two vectors draw the diagonal of the parallelogram starting from the tail of the two vectors:



Step 4:

Sketch the triangle half of the parallelogram and  use “law of cosines” and “law of sines” to determine magnitude and direction of  R:




Law of cosines:

Law of sines:

For problems and solutions on the parallelogram law, click HERE.

Glossary:

Parallel lines

Two or more lines that will NEVER meet each other.

Parallelogram

A quadrilateral (a four sided polygon) whose opposite sides are both parallel and equal in length.

For problems and solutions on the parallelogram law, click HERE.

Multiplication and Division of a Vector by a Scalar

The diagram below shows that when a vector is multiplied by a scalar, its magnitude changes but not its direction.  For example if vector A is multiplied by 2, its magnitude is doubled (i.e. length of the arrow doubles).  The same concept applies to the division of vectors.

Force Vectors - Scalars and Vectors

 

Scalars and Vector Table of Comparison