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Vectors

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Vectors In 3D Space

We define a #~{vector} (in 3-dimensional space) as a quantity with #~{magnitude} and #~{direction}.
Magnitude will be a positive real number. Direction is difficult to define without recourse to some arbitrary #~{frame of reference}, such as compas points in 2 dimensions, however the most important is the relative direction of two vectors to each other, which is expressed as the angle between them.

In general vectors will be denoted by lowercase latin letters in bold italic typeface, e.g. #~a, #~b, #~v, #~u etc. _ One special vector is the #~{null vector} denoted #0, which is the vector with zero magnitude and no direction.

Scalar Multiplication

Let #~a be a vector, we write | ~#a | for its magnitude. Let λ be a real number, we call λ a #~{scalar}, and we define #~{scalar multiplication} of ~#a , _ λ #~a _ as a vector which has magnitude | λ | | ~#a | , _ and direction the same as ~#a if λ is positive, the opposite direction if λ is negative.

Vector Addition

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If #~a and #~b are two vectors with an angle &theta. between them, then we define their sum as being the vector _ #~a + #~b _ given by the #~{parallelogram rule}:

#~a + #~b _ is the vector corresponding to the diagonal of the parallelogram defined by #~a and #~b (when they are joined at the origin) - see diagram left.

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As we can see on the diagram to the right, if &theta. is the angle between ~#a and ~#b then, by Pythagoras

_ _ _ | ~#a + #~b | ^2 _ = _ ( |~#a| sin &theta. ) ^2 + ( |#~b| + |#~a| cos &theta. ) ^2

_ _ _ _ _ _ _ _ = _ |~#a|^2 sin^2 &theta. + |#~b|^2 + 2 |#~a| |#~b| cos &theta. + |#~a|^2 cos^2 &theta.

_ _ _ _ _ _ _ _ = _ |~#a|^2 + |#~b|^2 + 2 |#~a| |#~b| cos &theta.

_ _ _ | ~#a + #~b | _ = _ &sqrt.${|~#a|&powtwo. + |#~b|&powtwo. + 2 |#~a| |#~b| cos &theta.}

and if &phi. is the angle between ~#a and #~a + ~#b then

_ _ _ &phi. _ = _ sin^{-1}( |~#a| sin &theta. ./ &sqrt.${|~#a|&powtwo. + |#~b|&powtwo. + 2 |#~a| |#~b| cos &theta. })

We have

  1. ~#a + #~b _ = _ #~b + #~a
  2. ~#a + #0 _ = _ #0 + #~a _ = _ #~a

I.e. the vectors form an Abelian group under addition.

 

 

Vector Algebra

With the above definitions of addition and scalar multiplication the following rules hold:

  1. λ(&mu. #~a) _ = _ (λ &mu.) #~a
  2. (λ + &mu.) #~a _ = _ λ #~a + &mu. #~a
  3. 1 #~a _ = _ #~a
  4. λ (#~a + #~b) _ = _ λ #~a + λ #~b
  5. λ #0 _ = _ #0 _ _ and _ _ 0 #~a = #0

Where #0, #~a and #~b are vectors and 1, 0, λ and &mu. are scalars.

In fact i)-iv) can be used as the definition of an , v) can be derived from i)-iv).

We can now define vector subtraction:

_ _ _ #~a - #~b _ #:= _ #~a + ({-1}) #~b

We write _ -#~b _ for _ ({-1}) #~b. _ This is the vector with the same magnitude as #~b, and the same direction, but opposite sense.