1. Multi-dimensional vectors play a far more critical role in electromagnetic theory than in simplemechanics, so some practice is helpful. Recall that there are two types of vector multiplication:the dot product and the cross product. The dot product of two vectors is a measure of how parallel they are; its magnitude is largestwhen the two vectors are parallel and smallest when they are perpendicular. Since the resultis just a number (not a vector), it is sometimes called the scalar product. For two vectors Aand B, the dot product is mathematically indicated as A · B. The value of the dot productis given by the formula AB cos e, in which A is the magnitude of A, B is the magnitude of Band 0 is the angle between A and B. (a) Using labeled arrows to represent two vectors A and B, make three sketches, one showingthe two vectors oriented to have a positive dot product, one showing an orientation givinga negative dot product, and one showing an orientation giving a zero value dot product. ) If you have the ryz components of two vectors, you can use this formula to find the dotproduct: A · B= AB# + A,By+ A¿Bz.

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