Vectors in a coordinate system: coordinates and magnitude

Coordinates of a vector

The vector AB, defined by two points A(xA; yA) and B(xB; yB), has the following coordinates:

vector AB (xB - xA; yB - yA)

Example: given A(1; 3) and B(4; -1), the vector AB (4–1; -1–3) = (3; -4).

Norm of a vector

The norm (length) of a vector u(x; y) is denoted by ||u|| and is calculated as follows:

||u|| = √(x² + y²)

Example: ||vector AB|| = √(3² + (-4)²) = √(9 + 16) = √(25) = 5. This is exactly the distance AB, which is to be expected since the norm of a vector AB is the length of the line segment [AB].

Addition and multiplication by a real number

If u(x; y) and v(x'; y'), then u + v has coordinates (x+x'; y+y'). Multiplying a vector by a real number k gives ku with coordinates (kx; k*y).

Equal vectors

Two vectors are equal if and only if they have exactly the same coordinates, even if their starting and ending points are different.

Pitfalls to avoid

• The vector AB goes from A to B: its coordinates are (xB – xA; y_B − y_A); we always start from the end point minus the start point.
• The vector BA is the opposite of the vector AB: vector BA = −vector AB.
• The magnitude of a vector is always positive or zero, never negative.