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I am havng difficulty understanding what to do for this problem. I know you have to take the cross product but how do you know what to cross product from the line. Thanks for any advice.

Consider the parallelogram of the following picture.

a) Rewrite the following cross products using (vector v) cross product (vector w). Simplify your answers.

i) (vector AB) cross product (vector AC)

ii) (vector AC) cross product (vector AD)

iii) (vector AB) cross product (vector DC)

iv) (vector AC) cross product (vector BD)

(b) For each of the following pairs of vectors, draw the original parallelogram and a new parallelogram whose sides are equal to the given vectors. Use these parallelograms to explain geometrically the corresponding algebraic result of (a).

i) vector AB, vector AC

ii) vector AC, vector AD

iv) vector AC, vector BD

For example for ai) I assume they just want you to write v x w? Then for aii from the picture you should know that AC=v+w so AC x AD = (v+w)xw=vxw since wxw =0.

So notice that you got the same answer for ai and aii. This is telling you two areas are equal since the length of the cross product can be interpreted as the area of a corresponding parallelogram.

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