3D Coordinates and Vectors

Nerd Cafe

1. Understanding 3D Coordinates

In 3D space, each point P(x, y, z) is represented by three values where:

• x is the position on the X-axis (left-right)

• y is the position on the Y-axis (front-back)

• z is the position on the Z-axis (up-down)

Example:

A point A(2, 3, 4) means:

• 2 units along X-axis

• 3 units along Y-axis

• 4 units along Z-axis

2. What is a Vector in 3D?

A vector in 3D space has:

• Direction

• Magnitude (length)

A 3D vector is written as:

$$\overrightarrow{v}=\left( a, b, c \right)$$

Where a, b, and c are the components along x, y, and z.

3. Vector from Two Points

To find a vector from point A to point B:

$$\overrightarrow{AB}=\left( x_{b}-x_{a} \; , y_{b}-y_{a} \; , z_{b}-z_{a} \right)$$

Example:

Let A = (1, 2, 3) and B = (4, 6, 9). Then

$$\overrightarrow{AB} =\left( 4−1,6−2,9−3 \right)=\left( 3,4,6 \right)$$

Python Code:

import numpy as np

A = np.array([1, 2, 3])
B = np.array([4, 6, 9])
vector_AB = B - A
print("Vector AB:", vector_AB)

Output:

Vector AB: [3 4 6]

2. Vector Magnitude (Length)

$$\left| \overrightarrow{v} \right|=\sqrt{x^{2}+y^{2}+z^{2}}$$

Example:

$$\overrightarrow{v}=(3,4,6)\Rightarrow \left| \overrightarrow{v} \right|=\sqrt{3^{2}+4^{2}+6^{2}}=\sqrt{61}$$

Python Code:

magnitude = np.linalg.norm(vector_AB)
print("Magnitude of Vector AB:", magnitude)

5. Unit Vector

A unit vector has magnitude 1 and points in the same direction:

$$\hat{v}=\frac{\overrightarrow{v}}{\left| \overrightarrow{v} \right|}$$

Python Code:

unit_vector = vector_AB / magnitude
print("Unit vector in direction of AB:", unit_vector)

Output:

Unit vector in direction of AB: [0.38411064 0.51214752 0.76822128]

6. Dot Product of Two Vectors

$$\overrightarrow{A}.\overrightarrow{B}=x_{a}x_{b}+y_{a}y_{b}+z_{a}z_{b}$$

Used to find the angle between vectors:

$$cos\theta=\frac{\overrightarrow{A}.\overrightarrow{B}}{|\overrightarrow{A}||\overrightarrow{B}|}$$

Python Code:

import numpy as np

A = np.array([1, 2, 3])
B = np.array([4, 5, 6])
dot_product = np.dot(A, B)
angle = np.arccos(dot_product / (np.linalg.norm(a) * np.linalg.norm(b)))
print("Dot product:", dot_product)
print("Angle (radians):", angle)

Output:

Dot product: 32
Angle (radians): 0.2257261285527342

7. Cross Product

$$\overrightarrow{A}\times \overrightarrow{B}=\begin{vmatrix} i & j & k \\ x_{a} & y_{a} & z_{a} \\ x_{b} & y_{b} & z_{b} \end{vmatrix}$$

Python Code:

import numpy as np

A = np.array([1, 2, 3])
B = np.array([4, 5, 6])

cross_product = np.cross(A, B)
print("Cross product (perpendicular vector):", cross_product)

Output:

Cross product (perpendicular vector): [-3  6 -3]

8. Distance Between Two Points in 3D

$$D=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}$$

Python Code:

import numpy as np

A = np.array([1, 2, 3])
B = np.array([4, 5, 6])

distance = np.linalg.norm(B - A)
print("Distance between A and B:", distance)

Output:

Distance between A and B: 5.196152422706632

9. Visualizing 3D Vectors

import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')

# origin
ax.quiver(0, 0, 0, a[0], a[1], a[2], color='blue', label='a')
ax.quiver(0, 0, 0, b[0], b[1], b[2], color='green', label='b')

ax.set_xlim([0, 5])
ax.set_ylim([0, 5])
ax.set_zlim([0, 5])
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('Z')
ax.legend()
plt.show()

Output:

Keywords

3D coordinates, vectors, vector magnitude, unit vector, dot product, cross product, vector projection, distance in 3D, 3D space, vector operations, python vectors, numpy vectors, angle between vectors, 3D visualization, vector direction, vector length, 3D math, linear algebra, vector geometry, vector representation, nerd cafe