Mathematics for Chemistry/Vectors

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Vectors
Imagine you make a rail journey from Doncaster to Bristol, from where you travel up the West of the country to Manchester. Here you stay a day, travelling the next morning to Glasgow, then across to Edinburgh. At the end of a day's work you return to Doncaster. Mathematically this journey could be represented by vectors, (in 2 dimensions because we are flat earthers on this scale). At the end of the 2nd journey (D-B) + (B-M) you are only a short distance from Doncaster, 50 miles at 9.15 on the clockface. Adding two more vectors, (journeys) takes you to Edinburgh, (about 250 miles at 12.00). Completing the journey leaves you at a zero vector away from Doncaster, i.e. all the vectors in this closed path add to zero.

Mathematically we usually use 3 dimensional vectors over the 3 Cartesian axes $$x$$, $$y$$ and $$z$$.

It is best always to use the conventional right handed axes even though the other way round is equally valid if used consistently. The wrong handed coordinates can occasionally be found erroneously in published research papers and text books. The memory trick is to think of a sheet of graph paper, $$x$$ is across as usual and $$y$$ up the paper. Positive $$z$$ then comes out of the paper.

A unit vector is a vector normalised, i.e. multiplied by a constant so that its value is 1. We have the unit vectors in the 3 dimensions:

$${\hat {\mathbf i}} + {\hat {\mathbf j}}  + {\hat {\mathbf k}}$$

so that

$${\mathbf v} = A_x{\hat {\mathbf i}}  + A_y {\hat {\mathbf j}}  + A_z {\hat {\mathbf k}}$$

The hat on the i, j, k signifies that it is a unit vector. This is usually omitted.

Our geographical analogy allows us to see the meaning of vector addition and subtraction. Vector products are less obvious and there are two definitions the scalar product and the vector product. These are different kinds of mathematical animal and have very different applications. A scalar product is an area and is therefore an ordinary number, a scalar. This has many useful trigonometrical features.

The vector product seems at first to be defined rather strangely but this definition maps onto nature as a very elegant way of describing angular momentum. The structure of Maxwell's Equations is such that this definition simplifies all kinds of mathematical descriptions of atomic / molecular structure and electricity and magnetism.

A summary of vectors
The unit vectors in the 3 Cartesian dimensions:

$${\hat {\mathbf i}} + {\hat {\mathbf j}}  + {\hat {\mathbf k}}$$

a vector $${\mathbf v}$$ is:

$${\mathbf v} = A_x{\hat {\mathbf i}}  + A_y {\hat {\mathbf j}}  + A_z {\hat {\mathbf k}}$$

The hat on the i, j, k signifies that it is a unit vector.

Vector magnitude
$$V_{mag}  =  \sqrt { { A_x }^2 +  { A_y }^2 +  { A_z }^2 }$$

A constant times a vector
$${\mathbf v_{new}} = c A_x {\hat {\mathbf i}}  + c A_y {\hat {\mathbf j}}  + c A_z {\hat {\mathbf k}}$$

Vector addition
$${\mathbf A}   + {\mathbf B}  = ( A_x + B_x  ) {\hat {\mathbf i}} + ( A_y + B_y  ) {\hat {\mathbf j}}  + ( A_z + B_z ) {\hat {\mathbf k}}$$

Notice $${\mathbf B}   + {\mathbf A} = {\mathbf  A}   + {\mathbf B}$$

Vector subtraction
$${\mathbf A}   - {\mathbf B}  = ( A_x - B_x  ) {\hat {\mathbf i}} + ( A_y - B_y  ) {\hat {\mathbf j}} + ( A_z - B_z ) {\hat {\mathbf k}}$$

Notice $${\mathbf A}   - {\mathbf B} = - ( - {\mathbf  A}   + {\mathbf B})$$

Scalar Product
$${\mathbf A}. {\mathbf B} =  A_x. B_x  +  A_y. B_y  +  A_z. B_z$$

Notice $${\mathbf B}. {\mathbf A} = {\mathbf A}. {\mathbf B}$$

Notice that if $${\mathbf A} = {\mathbf B} $$ this reduces to a square.

If A and B have no common non-zero components in $$x$$, $$y$$ and $$z$$ the value is zero corresponding to orthogonality, i.e. they are at right angles. (This can also occur by sign combinations making $${\mathbf A} . {\mathbf B} $$ zero. corresponding to non axis-hugging right angles.)

Vector product
$${\mathbf A}  {\rm x }{\mathbf B}  = ( A_y  B_z  - A_z B_y) {\hat {\mathbf i}} - ( A_x  B_z  - A_z B_x ) {\hat {\mathbf j}} +  ( A_x  B_y - A_y B_x) {\hat {\mathbf k}}$$

Notice $${\mathbf B}   {\rm x }{\mathbf A} = - {\mathbf  A}   {\rm x }{\mathbf B}$$

The minus sign on the middle term comes from the definition of the determinant, explained in the lecture. Determinants are defined that way so they correspond to right handed rotation. (If you remember our picture of $$\cos^2 + \sin^2 = 1$$ going round the circle, as one coordinate goes up, i.e. more positive, another must go down. Therefore rotation formulae must have both negative and positive terms.) Determinants are related to rotations and the solution of simultaneous equations. The solution of $$n$$ simultaneous equations can be recast in graphical form as a rotation to a unit vector in $$n$$-dimensional space so therefore the same mathematical structures apply to both space and simultaneous equations.