# 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:

Then appling cofactor expansion we rewrite the determinant opening it by the first row:

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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Then appling cofactor expansion we rewrite the determinant opening it by the first row:

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).

Dear Lyndsey

For you and other our visitors we created this video. Please take a look!

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