- #1
Taturana
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Could someone explain me the difference between the inner product and the dot product?
Thanks all
Thanks all
The inner product and dot product are two mathematical operations that are closely related. The main difference between them lies in the type of vectors they are applied to. The inner product is defined for both real and complex vectors, while the dot product is only defined for real vectors.
The inner product of two vectors a and b is calculated by multiplying the corresponding components of the vectors and then summing the results. This can be represented as a · b. The dot product, on the other hand, is calculated by taking the dot product of the two vectors and then dividing by the magnitude of the vectors. This can be represented as a · b = |a||b|cos(theta), where theta is the angle between the two vectors.
The inner product of two vectors can be thought of as a measure of how much the two vectors are pointing in the same direction. It is also related to the projection of one vector onto the other. The dot product, on the other hand, can be interpreted as the product of the magnitudes of the two vectors and the cosine of the angle between them. It is related to the concept of work and energy in physics.
The inner product and dot product have numerous applications in mathematics, physics, and engineering. They are used in vector calculus, linear algebra, and signal processing. In real life, they are used in areas such as computer graphics, image processing, and machine learning to calculate angles, distances, and projections between vectors.
Yes, both the inner product and dot product can be extended to higher dimensions. In fact, the inner product can be defined for any vector space, while the dot product can be defined for any Euclidean space. However, their properties and geometric interpretations may differ in higher dimensions.