3D Vectors

A 3D vector is a line segment in three-dimensional space running from point A (tail) to point B (head).

Each vector has a magnitude (or length) and direction.

3D Vector Space


Components of a Vector

If the coordinates of A and B are: A(x1, y1, z1) and B(x2, y2, z2) the coordinates or components of the vector Vector are the coordinates of the head minus the coordinates of the tail.

Components of a Vector

Calculate the components of the vectors that can be drawn in the triangle with vertices A(−3, 4, 0), B(3, 6, 3) and C(−1, 2, 1).

Triangle ABC

Vector

Vector

Vector


Magnitude or Length of a Vector

The magnitude of a vector is the length of the line segment that defines it.

The magnitude of a vector is always represented by a positive number and only the zero vector has a magnitude of zero.

Calculation of the Magnitude to Know Its Components

Vector

Module of a Vector

Given the vectors Vector and Vector, find the magnitudes of vector u and v·

Magnitude

Magnitude

Calculation of the Module Knowing the Coordinates of the Points

Vectors

Vector Modules


Distance between Two Points

Distance Between Two Points

Find the distance between the points A(1, 2, 3) and B(−1, 2, 0).

Vector Calculations


Unit Vector

The magnitude of the unit vector is one.

Normalizing a vector is obtaining another unit vector in the same direction.

To normalize a vector, divide it by its magnitude.

Unit Vector