Vector Calculation
Basics of vector calculation with examples
In the following descriptin are vector operations using vectors of length two or three. Vectors can have any number of entries.
Vector Addition
Vectors can only be added if the number of dimensions and their orientation (columns or roworiented) are the same.
The vectors \(\left[\matrix{X_a\\Y_a}\right] + \left[\matrix{X_b\\Y_b}\right]\) and \(\left[\matrix{X_a\\Y_a\\Z_a}\right] + \left[\matrix{X_b\\Y_b\\Z_b}\right]\) can be added.
The vectors \(\left[\matrix{X_a\\Y_a}\right] + \left[\matrix{X_b\\Y_b\\Z_b}\right]\) and \([X_a\;Y_a\;Z_a]+ \left[\matrix{X_b\\Y_b\\Z_b}\right]\) can not be added
Each number in the list is called a element. Vectors can be added and subtracted by adding and subtracting the elements.
\(\left[\matrix{a\\b}\right] + \left[\matrix{c\\d}\right] = \left[\matrix{a+c\\b+d}\right]\)
More infos about addition you can find here.
Vector Subtraction
The subtraction of vectors is identical to the addition of vectors, but with negative operator. For the vector subtraction, the same rules apply as for the addition of vectors.
More infos about subtraction you can find here.
Vector Multiplication
Vectors also can be multiplied by real numbers, as follows.
\(\left[\matrix{a\\b}\right]·\left[\matrix{c\\d}\right]=ac+bd\)
Note, the answer is a real number, not a vector.
More infos about multiplication and skalar product you can find here.
Scalar multiplication of a vector
Scalar vector multiplication is the multiplication of a vector with a real number. For this purpose, each element of the vector is multiplied by the real number.
\(a·\left[\matrix{x\\y\\z}\right]=\left[\matrix{a· x\\a·y\\a·z}\right]\)
\(5·\left[\matrix{2\\5\\4}\right]=\left[\matrix{5· 2\\5·5\\5·4}\right]=\left[\matrix{10\\25\\20}\right]\)
