3D Vectors (R³): Comprehensive Guide & Operations

What is a Vector?

A vector is a quantity that has both a magnitude and a direction. It is geometrically represented by a line segment whose length is the magnitude and an arrow that indicates its direction as shown in the figure below. Vectors are used in physics to model quantities with sizes and directions such as velocities, forces, accelerations; in engineering, chemistry, computer graphics, robotics, and many other fields.

In the figure above, the vector is defined using an initial point A and a terminal point B. Therefore the vector may be denoted as \( \vec{AB} \).

Equivalent Vectors

Vectors with equal magnitudes and the same direction are equivalent vectors, regardless of their position in space.

Geometric Vector Operations

Sum of two Vectors

Given two vectors \( \vec{v_1} \) and \( \vec{v_2} \), their sum is a vector obtained by first positioning vector \( \vec{v_2} \) such that its initial point coincides with the terminal point of \( \vec{v_1} \). The sum \( \vec{v_1} + \vec{v_2} \) is the vector whose initial point is the initial point of \( \vec{v_1} \) and its terminal point is the terminal point of \( \vec{v_2} \).

Note that \( \vec{v_1} + \vec{v_2} = \vec{v_2} + \vec{v_1} \). Also, the sum of two vectors coincides with the diagonal of the parallelogram determined by \( \vec{v_1} \) and \( \vec{v_2} \).

Difference of two Vectors

Given two vectors \( \vec{v_1} \) and \( \vec{v_2} \), the difference \( \vec{v_2} - \vec{v_1} \) may be defined as a sum \( \vec{v_2} + (- \vec{v_1}) \) and represented geometrically as shown below.

Multiplication of a Vector by a Scalar

A vector \( \vec{v_1} \) multiplied by a scalar \( k \) is defined as a vector \( k\vec{v_1} \) parallel to \( \vec{v_1} \) and whose direction is the same as that of \( \vec{v_1} \) if \( k \gt 0 \) and opposite if \( k \lt 0 \). The magnitude (length) of \( k\vec{v_1} \) is \( | k | \) times the magnitude of \( \vec{v_1} \).

The figure below shows vectors \( \vec{v_1} \), \( 2\vec{v_1} \) and \( -3 \vec{v_1} \).

Vectors in a 3D Rectangular Coordinate System

A unit vector is a vector with a magnitude equal to 1. Below is shown a 3D rectangular coordinate system with unit vectors \(\vec{i} \), \(\vec{j} \), and \(\vec{k} \) in the positive direction of the x, y, and z axes respectively. Vectors \(\vec{i} \), \(\vec{j} \), and \(\vec{k} \) may be defined by their components as follows:

• \(\vec{i} = \langle 1 , 0 , 0 \rangle\), one unit along the x axis.
• \(\vec{j} = \langle 0 , 1 , 0 \rangle\), one unit along the y axis.
• \(\vec{k} = \langle 0 , 0 , 1 \rangle\), one unit along the z axis.

Components of a Vector

The components of any vector \(\vec{v} \) are defined by expressing \(\vec{v} \) as a sum of multiples of the unit vectors \(\vec{i} \), \(\vec{j} \), and \(\vec{k} \) as follows:

Or in components form using angled brackets as follows:

The components of a vector \(\vec{v} \) defined by its initial point \( A = (x_1 , y_1 , z_1) \) and its terminal point \( B = (x_2 , y_2 , z_2) \) are given by:

Calculating Magnitude and Unit Vectors

Given vector \( \vec{v} = \langle a, b, c \rangle \), its magnitude (or length) is given by:

The unit vector \( \vec{u} \), defined as a vector of magnitude equal to 1, in the same direction as \( \vec{v} \) is given by:

Algebraic Sum, Difference, and Scalar Multiplication

Given vectors \( \vec{v_1} = \langle a_1, b_1, c_1 \rangle \) and \( \vec{v_2} = \langle a_2, b_2, c_2 \rangle \), the sum \( \vec{v_1} + \vec{v_2} \), the difference \( \vec{v_1} - \vec{v_2} \), and scalar multiplication \( k \vec{v_1} \) (where k is a real number) are given by: