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Answer to Question #5959 in Linear Algebra for Lyndsey

Question #5959
find (1,0,0) x (0,0,1)
Expert's answer
In this task we need to calculate vector product(or cross product) of two vectors a=(1,0,0) and b=(0,0,1). Recall the definition, if a={a1,a2,a3} and b={b1,b2,b3} then cross product is a vector which is perpendicular to both vectors a={a1,a2,a3} and b={b1,b2,b3} and it’s calculated in the following way. We construct determinant out of coordinates of given vectors a and b as shown below:
data:image/png;base64,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
Then appling cofactor expansion we rewrite the determinant opening it by the first row:
data:image/png;base64,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
So axb=(a2*b3 - a3*b2,a3*b1 - a1*b3,a1*b2 - a2*b1).

Hence if a=(1,0,0) and b=(0,0,1) then we have
(1,0,0)x(0,0,1)=(0*1-0*0,-(1*1-0*0),1*0-0*0)=(0,-1,0).


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