1.2. Vectors

Vectors in 2D and 3D space

Vectors

Vector and points are both lists of three numbers so there’s absolutely no way of telling whether it is a point or a vector.

A vector is an arrow in space which always starts at the world origin (0, 0, 0) and ends at the specified coordinate.

A vector is a quantity that describes Direction and Magnitude.

Vector 2d

2D Vectors

A 2D vector can be specified in the following way as a matrix:

$$\overrightarrow{v} = \begin{bmatrix} x_1 \\
y_2 \end{bmatrix}$$

$$\overrightarrow{v} = \begin{bmatrix} 2 \\
3 \end{bmatrix}$$

or as a list:

$$\overrightarrow{v} = \begin{bmatrix} 2, 3 \end{bmatrix}$$

Vector 2d

3D Vectors

A 3D vector can be specified in this way as a matrix:

$$\overrightarrow{v} = \begin{bmatrix} x_1 \\
y_1 \\
z_1 \end{bmatrix}$$

$$\overrightarrow{v} = \begin{bmatrix} 2 \\
3 \\
3 \end{bmatrix}$$

or as a list:

$$\overrightarrow{v} = \begin{bmatrix} 2, 3, 3 \end{bmatrix}$$

Vectors

Operations with vectors

Addition

The sum of two vectors is a third vector, represented as the diagonal of the parallelogram constructed with the two original vectors as sides.

$$\overrightarrow{AB} = \begin{bmatrix} a1 \\
a2 \\
a3 \end{bmatrix} +\begin{bmatrix} b1 \\
b2 \\
b3 \end{bmatrix} = \begin{bmatrix} a1+b1 \\
a2+b2 \\
a3+b3 \end{bmatrix} $$

Vectors addition

Multiplication

When a vector is multiplied by a positive scalar (i.e., number), its magnitude is multiplied by the scalar and its direction remains unchanged (if the scalar is negative, the direction is reversed).

$$\overrightarrow{v} = \begin{bmatrix} a1 \\
a2 \\
a3 \end{bmatrix} 3 = \begin{bmatrix} a1 \cdot 3 \\
a2 \cdot 3 \\
a3 \cdot 3
\end{bmatrix} $$

Vectors addition

Vector multiplication: The vector is now half its original size (multiplied by 0.5).

Magnitude

The magnitude of a vector is its size. It can be calculated from the square root of the total of the squares of of the individual vector components.

Normalization

To normalize a vector, therefore, is to take a vector of any length and, change its length to unit 1, turning it into what is called a unit vector. Since it describes a vector’s direction without regard to its length, it’s useful to have the unit vector readily accessible.

Dot product

The multiplication of a vector A by another vector B leads to the dot product (also known as scalar product), returning a number.

The dot product of two vectors can be defined as the product of the magnitudes of the two vectors and the cosine of the angle between the two vectors.

Alternatively, it is defined as the product of the projection of the first vector onto the second vector and the magnitude of the second vector.

$$ A \cdot B = |A| |B| cos (\alpha)$$

Dot Product Dot Product

Cross product

Also called the vector product of two vectors A (Red) and B (Green), resulting a third vector C (Blue), perpendicular (Normal Vector) to the plane of the original vectors (A and B). Its magnitude is given by the area of the parallelogram between A and B, and its direction can be determined by the right-hand thumb rule (See bellow).

$$ A \times B = |A| |B| sin (\alpha) $$

Cross Product

Right-hand rule

Right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. This will help you in remembering the direction of the third vector resulting from a Cross Product multiplication of two vectors.

Right hand rule

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