Talking about dot products in linear algebra can be really a subject of hours, but long story short this is probably the minimal knowledge you can live with:
- The dot product (inner product) is an operation on 2 vectors that return a scalar.
E.g., given 2 vectors u and v, have their result np.dot(u, v) to be a real number. - The dot product works on vectors in 2D, 3D or other dimensions.
- The dot product is measuring “how aligned” 2 vectors are.
- In 2D space, parallel vectors, pointing in the same direction, have dot products of 1.
- In 2D space, parallel vectors, pointing in the opposite directions, have dot products of -1.
- Perpendicular vectors, forming a 90-degree angle, always produce a dot product of 0 due to the cosine of the angle being 0.
- For vectors with arbitrary angles between them, the dot product varies according to the cosine of the angle formed, resulting in values other than 0, 1, or -1.
Pretty much that is just the top of the iceberg, but it is good to illustrate this a bit:
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import numpy as np import matplotlib.pyplot as plt v1 = np.array([1, 0]) v2 = np.array([0, 1]) v3 = np.array([0.6, 0.8]) v31 = np.array([0.8, -0.6]) v4 = np.array([0.6, -0.8]) v5 = np.array([1, 0]) v6 = np.array([-1, 0]) # Plotting plt.figure(figsize=(5, 5)) # v1, v2, v3, v31, v4, v5, v6 for i, v in enumerate([v3, v31, v4]): plt.quiver(0, 0, v[0], v[1], angles='xy', scale_units='xy', scale=1, color='C'+str(i), label='vector' + str(i+1)) plt.xlim(-2, 2) plt.ylim(-2, 2) plt.axhline(0, color='black',linewidth=0.7) plt.axvline(0, color='black',linewidth=0.7) plt.grid(color = 'gray', linestyle = '--', linewidth = 0.5) plt.title('Dot Products in Linear Algebra') plt.legend() plt.show() |
Returns the following, where it is obvious which vectors are perpendicular:
Vector 2 and Vector 1 are perpendicular, with degree between them equalling 90 and a dot product of 0.
Thank you for your interest and enjoy the linear algebra and your day!