11 Vectors
Vectors are the mathematical tools needed when we deal with engineering systems in
more than one dimension. Vector quantities such as force have both a magnitude and
direction associated with them, and so are represented by mathematical symbols that,
similarly, have a magnitude and direction. We can construct an algebra of such quantities by
which they may be added, subtracted, multiplied (in two different ways) – but not divided.
These rules of combination reflect the ways in which the corresponding physical quantities
behave – for example vectors add in the same way that forces do, the scalar product
represents ‘vector times vector=scalar’ in the same way that ‘velocity times velocity=
energy’. By introducing a coordinate system, we can also represent vectors by arrays of
ordinary numbers, in which form they are easier to combine and manipulate.
Prerequisites
It will be useful if you know something about- ratio and proportion (14
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) - Cartesian coordinate systems (205
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) - Pythagoras’ theorem (154
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) - sines and cosines (175
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) - cosine rule (197
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)- simultaneous linear equations (48
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)- function notation (90
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)- differentiation (Chapter 8
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)Objectives
In this chapter you will find- definitions of scalars and vectors
- representation of a vector
- addition and subtraction of vectors
- multiplication of a vector by a scalar
- rectangular Cartesian coordinates in three dimensions
- distance in Cartesian coordinates
- direction cosines and ratios
- angle between two lines
- basis vectors (i,j,k)