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The dot product, also known as the scalar product, has a geometric interpretation related to the concept of orthogonality and the angle between vectors.
The dot product of two vectors, denoted as A · B, is defined as the product of their magnitudes and the cosine of the angle between them. Geometrically, it represents the projection of one vector onto another, multiplied by the magnitude of the other vector.
Here are a few examples to illustrate the geometric interpretation of the dot product:
1. Orthogonal Vectors: If the dot product of two vectors is zero (A · B = 0), it indicates that the vectors are orthogonal or perpendicular to each other.
For example, in a 2D Cartesian coordinate system, the vectors (1, 0) and (0, 1) are orthogonal because their dot product is zero.
2. Parallel Vectors: When two vectors are parallel, their dot product is equal to the product of their magnitudes.
For instance, if vector A = (2, 3) and vector B = (4, 6), their dot product is $[2*4 + 3*6] = 20$, which is equal to the product of their magnitudes $\sqrt {(2^2 + 3^2)} * \sqrt{(4^2 + 6^2)} = 20$.
3. Angle between Vectors: The dot product can be used to find the angle between two vectors. By rearranging the formula A · B = |A| |B| cos(theta), we can solve for the angle theta.
For example, if vector A = (3, 4) and vector B = (1, 2), their dot product is $[3*1 + 4*2]= 11$. By using the formula, we can find the angle between them as $cos(\theta)=\frac{11}{\sqrt{(3^2 + 4^2)} * \sqrt{(1^2 + 2^2)}}$.
