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 row-oriented) 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.


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.

\(a·\left[\matrix{x\\y\\z}\right]=\left[\matrix{a· x\\a·y\\a·z}\right]\)
\(5·\left[\matrix{2\\5\\4}\right]=\left[\matrix{5· 2\\5·5\\5·4}\right]=\left[\matrix{10\\25\\20}\right]\)

Vector Definition
Vector Calculation
Vector Addition
Vector Subtraction
Vector Magnitude
Vector Dot Product
Vector Cross Produkt
Vector Angle
Triple Product

Is this page helpful?            
Thank you for your feedback!

Sorry about that

How can we improve it?