RSS feed for Introducing vectors for engineering applications

Introduction

The Open University — Mon, 25 Feb 2019 00:00:00 GMT

Applied mathematics is a key skill for practicing engineers and mathematical modelling is an ever-increasing field within engineering. .

In Section 1 you will explore how vectors are used to model force and motion, and consider how problems involving vectors can be solved using geometry and trigonometry. In Section 2 you explore how to work with vectors represented in component form. Section 3 is concerned with vector algebra, and considers how equations involving vectors can be solved. Finally, Section 4 introduces the scalar product of vectors, a multiplication operation that takes into account direction as well as magnitude.

Solutions to the activities which appear in this course can be found on this page.

This OpenLearn course is an adapted extract from the Open University course T194 Engineering: mathematics, modelling, applications

Learning outcomes

The Open University — Mon, 25 Feb 2019 00:00:00 GMT

After studying this course, you should be able to:

identify if a quantity is a vector
represent vectors from engineering problems in an appropriate form
model simple engineering systems (such as combining forces) using vectors
perform simple algebraic procedures using vectors.

Background

The Open University — Mon, 25 Feb 2019 00:00:00 GMT

It is common in engineering for physical phenomena to be represented as vector fields. A vector field is a mathematical representation of a system that describes how a quantity, such as a force, changes over an interval of time, or an area or volume of space. Figure 1, for example, illustrates vector fields created by magnets (in part (a)) and fluid flow (in part (b)). A vector field can be thought of as a graph where every coordinate has not only a position, but also a magnitude and direction. In the images in Figure 1, these are represented by small arrows, with each individual arrow indicating the direction and magnitude of a force at a specific position.

View larger image

Figure 1 Examples of models of vector fields: (a) model of a vector field created by a bar magnet; (b) model of a vector field created by flow around a cylinder

Object	Approximate magnitude of acceleration (m s⁻²)
High-speed train	0.25
Executive car	4.3
Sprinter (pulling away from start line)	9.2
Gravity on Earth at sea-level standard	9.8
Saturn V moon rocket (just after launch)	11.2
Mid-engined sports car	15.2
Space shuttle (maximum during launch)	29
Formula One car (maximum under heavy braking)	49
F-16 aircraft (pulling out of dive)	79
Explosive seat ejection from aircraft	147
Automobile crash (100 km h⁻¹ into wall)	982
Football struck by foot	2946
Baseball struck by bat	29 460
Closing jaws of a trap-jaw ant	1 000 000
Jellyfish stinger	53 000 000

RSS feed for Introducing vectors for engineering applications

Introduction

Learning outcomes

Background

1 Modelling with vectors

1.1 Modelling motion with perpendicular vectors

1.1.2 Calculating the magnitude of a combined force

Activity 1

1.1.3 Calculating direction of motion of a combined force

Opposite, corresponding and alternate angles

Activity 2

1.1.4 Calculating acceleration from a combined force

Example 1 Finding acceleration from a combined force

Solution

1.2 Models of motion

Table 1 Approximate magnitudes of acceleration

Equations of linear motion

Example 2 Calculating speed and distance

Solution

Activity 3

How good is the model?

1.3 Modelling motion with non-perpendicular vectors

Activity 4

Activity 5

Activity 6

2 Vectors in component form

2.1 Horizontal and vertical components

Activity 7

2.2 Cartesian unit vectors

Component form of a vector

Activity 8

Alternative component form of a vector

Activity 9

2.3 Converting between vector forms

Component form of a vector in terms of its magnitude and angle with the positive x-axis

Activity 10

Magnitude and direction of a vector in terms of its components

Activity 11

2.4 Position vectors

Activity 12

3 Vector algebra with components

3.1 Vector addition in component form

Activity 13

Adding vectors in component form

Adding column vectors in component form

Activity 14

Example 3 Calculating the magnitude of combined forces

Solution

3.2 Scalar multiplication of vectors in component form

Scalar multiplication of a vector in component form

Activity 15

3.3 Vector subtraction in component form

Subtracting vectors in component form

Example 4 Calculating vector subtraction in component form

Solution

Activity 16

3.4 Combining vector operations

Example 5 Simplifying a combination of vectors in component form

Solution

Activity 17

3.5 Vector algebra

Properties of vector algebra

Example 6 Simplifying vector expressions

Solution

Activity 18

4 Scalar product of vectors

4.1 Scalar product of a vector from components

Scalar product of vectors in terms of components

Activity 19

4.2 Scalar product of a vector from magnitude and direction

Scalar product of vectors in terms of magnitude and direction

Activity 20

4.3 Properties of the scalar product

Scalar product properties

Example 7 Simplifying an expression containing a scalar product

Solution

Activity 21

4.4 Finding the angle between two vectors

Angle between two vectors

Example 8 Calculating the angle between two vectors in component form