Find the cross product a X b and verify that it is orthogonal to both a and b.

Brennan Flores 2021-09-25 Answered

Find the cross product a×b and verify that it is orthogonal to both a and b.
a=(2,3,0),b=(1,0,5)

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Expert Answer

Aamina Herring
Answered 2021-09-26 Author has 85 answers

Solution:
To find the cross product of two vectors a and b, where the components of both vectors are as follows
a=<a1,a2,a3>
b=<b1,b2,3>
We use the following formula,
c=a×b
=|i^j^k^a1a2a3b1b2b3|
=i^|a2a3b2b3|j^|a1a3b1b3|+k^|a1a2b1b2|
=i^(a2b3a3b2)j^(a1b3a3b1)+k^(a1b2a2b1)
In order to prove that this vector is orthogonal to both vectors a and b, we need to show that the dot product of vector c to each vector is zero, i.e we need to show that
ca=0
c b=0
Calculations:
We use equation, in order to find vector c which is the cross product of the two vectors a and b as follows
c=a×b
=|i^j^k^230105|
=i^|3005|j^|2015|+k^|2310|
=i^((3)(5)(0)(0))j^((2)(5)(0)(1))+k^((2)(0)(3)(1))
=15i^10j^3k^
Thus, the cross product of the two vector is the following vector
c=<15,10,3>
We find the dot product of vector c with vector a, and then b in order to find out whether if its orthogonal to both or now
ca=<15,10,3><2,3,0>
=3030+0
=0 (1)
And, the dot product of the vector c and b
ca=<15,10,3><1,

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