How to use in-sentence of “dot product”:
– Flux is the dot product of E and dA.
– Given a vector field is the dot product operation.
– This property of the dot product has several useful applications.
– Moreover, two vectors can be considered orthogonal if and only if their dot product is zero, and they both have a nonzero length.
– If both a and b are unit vectors, then their dot product simply gives the cosine of the angle between them.
– For vectors with complex entries, using the given definition of the dot product would lead to quite different geometric properties.
– In other words, for an orthonormal space with any number of dimensions, the dot product is invariant under a coordinate transformation based on an orthogonal matrix.
Example sentences of “dot product”:
– This definition naturally reduces to the standard vector dot product when applied to vectors, and matrix multiplication when applied to matrices.
– Based on this reasoning, to find the flux, we need to take the dot product of v with the unit surface normal to “S” at each point, which will give us a scalar field, and integrate the obtained field as above.
– If only b is a unit vector, then the dot product a · b givesa cos.
– For instance, the dot product of a vector with itself can be an arbitrary complex number, and can be zero without the vector being the zero vector; this in turn would have severe consequences for notions like length and angle.
– The same way, in a dimension 3, the dot product of vectors and is ad + be + cf.
– In mathematics, the dot product is an operation that takes two vectors as input, and that returns a scalar number as output.
– Occasionally, a double dot product is used to represent multiplying and summing across two indices.
– The double dot product between two 2nd order tensors is a scalar.
– The dot product between a tensor of order n and a tensor of order m is a tensor of order n+m-2.
– The dot product is one method to multiply vectors.
– The dot product is also a scalar in this sense, given by the formula, independent of the coordinate system.
– As the cosine of 90° is zero, the dot product of two orthogonal vectors is always zero.
– The Frobenius inner product generalizes the dot product to matrices.
– The dot product is worked out by multiplying and summing across a single index in both tensors.
- This definition naturally reduces to the standard vector dot product when applied to vectors, and matrix multiplication when applied to matrices.
- Based on this reasoning, to find the flux, we need to take the dot product of v with the unit surface normal to "S" at each point, which will give us a scalar field, and integrate the obtained field as above.