Multi-dimensional vectors play a far more critical role in electromagnetic theor

Question

Multi-dimensional vectors play a far more critical role in electromagnetic theory than in simple mechanics, 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 largest when the two vectors are parallel and smallest when they are perpendicular. Since the result is just a number (not a vector), it is sometimes called the scalar product. For two vectors A and B, the dot product is mathematically indicated as A middot B. The value of the dot product is given by the formula AB cos theta, in which A is the magnitude of A, B is the magnitude of B and theta is the angle between A and B. (a) Using labeled arrows to represent two vectors A and B, make three sketches, one showing the two vectors oriented to have a positive dot product, one showing an orientation giving a negative dot product, and one showing an orientation giving a zero value dot product. (b) If you have the xyz components of two vectors, you can use this formula to find the dot product: A middot B = A_xB_x + A_yB_y + A_z B_z For the vectors A = 5i + 4j + 2k and B = 2i + 1j + 2k, determine the dot product A middot B. After you have found the dot product, find the angle between the two vectors A and B.

Explanation / Answer

b) A = 5i + 4j + 2k

B = 2i + j + 2k

A.B = AxBx + AyBy + AzBz

= 5*2 + 4*1 + 2*2

= 18

so the dot product is 18

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