Cross Product of Vectors in the Plane Let = a1 + a2 and = b1 + b2 be two nonpara

Question

Cross Product of Vectors in the Plane Let = a1 + a2 and = b1 + b2 be two nonparallel vectors in 2-space, as in Figure 13.49. Figure 13.49 Use the identity sin (beta - alpha) = (sin beta cos alpha - cos beta sin alpha) to derive the formula for the area of the parallelogram formed by and :Area of parallelogram = |a1b2 - a2b1|. Show that a1b2 - a2b1 is positive when the rotation from to is counterclockwise, and negative when it is clockwise. Use parts (a) and (b) to show the geometric and algebraic definitions of times give the same result. The Dot Product in Genetics2 We define3 the angle between two n-dimensional vectors, and , using the dot product: cos theta = ,provided , 0.

Explanation / Answer

a) area of parllelogram

|a1b2 - a2b1|

= r cos s sin - r sin s cos

= rs ( sin cos - cos sin)

= rs sin( - )

thus answer.

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