Dot and cross product represent two \"types\" of vector multiplication. Discuss

Question

Dot and cross product represent two "types" of vector multiplication. Discuss their fundamental differences and similarities. There is considerable trigonometry involved in each case; this may help you see the differences (and similarities) between the two operations. Explain, for example, why you can cross three vectors (two at a time, following the usual rules), but not dot three vectors. Do you see the dot product "in action" in matrix multiplication? What sort of insights can the dot product give when trying to comprehend matrix multiplication?

Explanation / Answer

The "dot" or "inner" product of two vectors is the sum of the vector component products. This can be represented as a matrix multiplication. row vector (1x3) *column vector (3x1) = scalar (1x1). The "cross" or "outter" product is related to the vector product when you reverse the order of the matrix multiplication. Because the cross product result must map back to a 3-D space (vector), the unit vector cross products are defined in the following way: i x i = 0, j x j = 0, k x k = 0. i x j = k, j x k = i, k x i = j. i x k = -j, k xj = -i, j x i = -k. column vector (3x1)*row vector (1x3) = matrix (3x3). To get the cross product vector, the 3x3 matrix entries keeping track of the crossed vector component terms are summed. This results in a vector that is orthogonal to both the output vectors, because the cross product of the unit vectors are defined to be orthogonal. There is a short cut for finding the cross product, and this involves calculating a determinate of a 3x3 matrix with rows of the component vector, first vector and second vector. The short-cut is all most textbooks (and instructors) will show you, and it doesn't really help with understanding what the cross product means. Now, back to your question. The dot product is used for finding the magnitude of the part of a vector in a given direction. One form of work is done when a force is applied over a given distance. The only force that actually does work is the one that results in movement. So to calculate the work, you are only interested in the part of the force vector that is in the same direction as the displacement. To calculate the work done by a force that causes a rotation needs to use the cross product. The work in this case is the dot product of a torque vector, which is the cross product of the force and radius vectors, and the rotation vector, which points along the axis of rotation, and the magnitude is how far the object rotates. You can't calculate work by using a cross product, because the cross product produces a vector that is orthogonal to the two input vectors. Work is a scalar in that it represents the magnitude of a force in a given direction times the movement in that direction.

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