Basic vector mathematics

In this article I will talk about basic vector operations and their uses. Vectors are definitely extremely useful in multivariable calculus and actually without them it would be impossible to go further in study of multivariable calculus. If we start at the beginning a vector is an object not having only scalar value, but also a direction. One of the basic and useful vector operations is the dot product. It is a sum of multiplied nth member of each vector with the other vector's nth member. Here is the symbolical notation with the actual operation of dot product.

Importantly the result of a dot product is a scalar! The sum of multiplied vectors by each other is the algebraic look at the dot product. If we look at it geometrically we can get even more useful formula. Geometrically we can look at the dot product as if two vector would create a triangle with the third side being their difference.

Through the law of cosine:

source: http://en.wikipedia.org/wiki/Law_of_cosines?

we can write this equation resulting the new formula. It is based on the triangle above.

So we get a useful equation and formula for calculating angles, lengths through the dot product. From this equation we can calculate cosine of angle theta and results this formula. Therefore:

This is very important formula for later on proving perpendicular angles (e.g. proof that gradients of functions are perpendicular to the function at that point and level curve.) If the dot product of two vectors equals zero, they are perpendicular to one another. 

Another important operation with vectors is the cross product. By its name it uses symbol "x". Even though these operations are called products, that doesn't mean there are multiplications. Result of a cross product operation is a vector, unlike the dot product. Importantly the vector is perpendicular to both the those multiplied ones (therefore cross product makes sense only in space and vectors with three dimensions). The result can be calculated through determinants of vectors (actually matrix with vectors in it). Determinants of matrices are used to e.g. create an inverse matrix, to solve system of linear equations later on.

"In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well. The determinant provides important information about a matrix of coefficients of a system of linear equations, or about a matrix that corresponds to a linear transformation of a vector space."
source: http://en.wikipedia.org/wiki/Determinant

So, first of all matrices are labeled this way.

We can imagine two vectors ((a, b), (c, d)) being in the matrix M. If we want to compute the determinant of this matrix it's:
det(M) = a * d - b * c

But as we said earlier, cross product is a 3-dimensional vectors operation, so there are three columns in the matrix. Then the determinant of 3x3 (we will se why 3x3 matrix) is calculated through the Laplace expansion. So lets have two space vectors.

The rights side of the equation is another way of labeling vectors - through the unit vectors. Because vectors are amount of displacements, we can have i being the unit vector (1, 0) and multiply it by a1 (for example vector (3, 4, 5) = 3*i + 4*j + 5*k) and then when added together with other unit vectors j and k, the overall displacement results the vector.

This is how will the final cross product (determinant of 3x3 matrix) look like with the Laplace expansion.

Signs by which individual part of the resulting expansion are multiplied by are very important. Because Laplace expansion is based on calculating the ij-minors and ij-cofactors of matrix M and individual members (a1, a2 e.g.). The ij-cofactor of matrix M can be calculated with this equation:

Resulting a "checkerboard" like matrix, where each individual is in the Laplace expansion multiplied by the (-1) ^ (i + j).

So, now we now how to calculate the cross product and determinants. One of the uses of cross product is to find a vector perpendicular to two other vectors. Other use is that the magnitude of a cross product results an area of a parallelogram.

We see the magnitude of a cross product (vector product sometimes mentioned) results the area of a parallelogram and the cross product itself result another vector perpendicular to both of them.

These are the two "main" basic vector operations. I am sorry for self-drawn pictures, and hopefully all of these explanations were understandable, I think it is important to know why certain operations work and how to actually compute them, without having to remember a formula.

Lukas Cerny, 5. 7. 2014