Definition of the Dot Product
The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers (vectors) and returns a single number (a scalar). Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them, quantifying how much two vectors point in the same direction.
How to Calculate the Dot Product
For two vectors, A = [A₁, A₂, ..., An] and B = [B₁, B₂, ..., Bn], the dot product is calculated as the sum of the products of their corresponding components: A · B = A₁B₁ + A₂B₂ + ... + AnBn. Alternatively, using the magnitudes |A| and |B| and the angle θ between them, it's A · B = |A| |B| cos(θ).
Practical Example
Consider two 2D vectors: A = [3, 4] and B = [5, 2]. The dot product A · B = (3 * 5) + (4 * 2) = 15 + 8 = 23. If A = [2, 0] (pointing along the x-axis) and B = [0, 3] (pointing along the y-axis), their dot product is (2*0) + (0*3) = 0, indicating they are perpendicular.
Importance and Applications
The dot product is crucial in physics for calculating work (force · displacement), magnetic flux (magnetic field · area vector), and electric potential. In geometry, it is used to determine the angle between vectors, check for orthogonality (perpendicularity if the dot product is zero), and find vector projections.