Coordinates and operations on vectors in space

Positioning in space

In a space with a coordinate system (O, i, j, k), any point M is identified by its coordinates (x, y, z). A vector u = vector(AB), associated with the points A(xA, yA, zA) and B(xB, yB, zB), has the following coordinates: u(xB - xA, yB - yA, zB - zA)

Norm of a vector

The norm (length) of a vector u(x, y, z) is given by: ||u|| = √(x² + y² + z²)

Example: if u(2, -1, 2), then ||u|| = √(4 + 1 + 4) = √9 = 3.

Operations on vectors

As in the plane, we define:

• Sum: u + v = (x₁ + x₂, y₁ + y₂, z₁ + z₂)
• Scalar product: k·u = (k·x, k·y, k·z)

Operation	Formula	Example with u(1,2,-1), v(3,0,2)
u + v	(x₁ + x₂, y₁ + y₂, z₁ + z₂)	(4, 2, 1)
2u	(2x₁, 2y₁, 2z₁)	(2, 4, -2)
u - v	(x1-x2, y1-y2, z1-z2)	(-2, 2, -3)

Common pitfall

Be careful not to confuse the coordinates of a point with those of a vector: vector(AB) is ALWAYS calculated as ‘coordinates of B minus coordinates of A’, never the other way round. Reversing the order changes the direction (and the sign) of the vector.