The dot product is a scalar that is equal to the length of the projection of one vector X onto a unit vector pointing in the same direction as another vector Y, provided that X and Y start in the same location. The cross product is a vector that is perpendicular to two other vectors X and Y.
Know MoreThe dot product is calculated by multiplying together the first element, the second element and so on of X and Y; these products are summed to yield the dot product. The dot product is also found by multiplying together the length of X, the length of Y and the cosine of the angle between X and Y. Calculating the cross product requires finding the determinant of a matrix in which the first row contains unit vectors, the second row contains the vector X and the third row contains the vector Y. Although the dot product is commutative (X dot Y is the same quantity as Y dot X), the cross product is not. X cross Y is the determinant of the matrix described above, while Y cross X is the determinant of the matrix where the first row contains unit vectors, Y is the second row and X is the third row.
Learn more about CalculusA good application for vector analysis is the calculation of an airplane's flight path. This requires knowing the magnitude and direction of both the plane's thrust, as well as that of the prevailing winds acting on the plane.
Full Answer >Lamar University explains that a gradient vector field is a directional derivative that represents the change in the gradient of a function. Gradient fields are known as conservative fields since they depend only on the beginning point and the end point of the vector while ignoring the path taken by the vector. Another term for gradient vector fields is path-independent vectors.
Full Answer >Logarthims of the same base can be added together by multiplying their arguments and then performing the logarithm on the product. For example, assuming log means log base 10 as it does on a calculator: log(x) + log(y) = log(x * y)
Full Answer >Orthogonal vectors, also known as perpendicular vectors, are a type of vectors whose dot product or scaler is zero. However, if their cross product is zero, the vectors are said to be parallel.
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