Dot product and cross product

In summary, the conversation discusses the reasoning behind finding the product of a dot product and a vector, as well as the confusion around adding this product to another expression. The conversation concludes by clarifying that, in this scenario, the dot product between orthonormal vectors is always equal to zero and that the cross product of two vectors can be determined by using the third vector in the orthonormal basis.
  • #1
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Homework Statement
Let B = (v⃗1, v⃗2, v⃗3) be any basis of R3 consisting of perpendicular unit vectors, such that v⃗3 = v⃗1 × v⃗2. Let T(x⃗) = v⃗1 × x⃗ + (v⃗1 · x⃗)v⃗1. Find the B-matrix B of the given linear transformation T from R3 to R3. Interpret T geometrically.
Relevant Equations
dot product
cross product
1667972360004.png

Could anyone explain the reasoning from step 2 to step 3?

Specifically, I don't understand how to find the product of a cross product and a vector - like (v1 · v2)v1 and (v1 · v3)v1. I'm also confused by v1 × v3 + (v1 · v3)v1 -- is v1 × v3 = v1v3? How would this be added to (v1 · v3)v1?

Thank you.
 
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  • #2
jolly_math said:
Specifically, I don't understand how to find the product of a cross product and a vector - like (v1 · v2)v1 and (v1 · v3)v1.
There is no cross product and a vector. There is a dot product and a vector. The dot product is just a scalar multiplying the vector.

jolly_math said:
I'm also confused by v1 × v3 + (v1 · v3)v1 -- is v1 × v3 = v1v3? How would this be added to (v1 · v3)v1?
The vectors are orthonormal so ##\vec v_1\cdot \vec v_3=0##. The expression therefore reduces to ##\vec v_1\times\vec v_3##.
 
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  • #3
Orodruin said:
There is no cross product and a vector. There is a dot product and a vector. The dot product is just a scalar multiplying the vector.
For the second transformation, v1 x v2 = v3, but what does (v1 · v2)v1 equal?
Orodruin said:
The vectors are orthonormal so ##\vec v_1\cdot \vec v_3=0##. The expression therefore reduces to ##\vec v_1\times\vec v_3##.
Why would ##\vec v_1\times\vec v_3## = ##-\vec v_2##?

Thanks.
 
  • #4
jolly_math said:
but what does (v1 · v2)v1 equal?
What is ##\vec v_1\cdot \vec v_2## if all ##\vec v_i## are orthonormal?

jolly_math said:
Why would v→1×v→3 = −v→2?
Because you know that the ##\vec v_i## form an orthonormal basis and that ##\vec v_1\times\vec v_2=\vec v_3##.
 
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  • #5
Orodruin said:
What is ##\vec v_1\cdot \vec v_2## if all ##\vec v_i## are orthonormal?
The dot product would be zero, I understand the second transformation now.

Orodruin said:
Because you know that the ##\vec v_i## form an orthonormal basis and that ##\vec v_1\times\vec v_2=\vec v_3##.
I don't have much experience with cross products, does ##\vec v_1\times\vec v_2=\vec v_3## directly lead to ##\vec v_1\times\vec v_3 = -\vec v_2##?
 
  • #6
jolly_math said:
I don't have much experience with cross products, does ##\vec v_1\times\vec v_2=\vec v_3## directly lead to ##\vec v_1\times\vec v_3 = -\vec v_2##?
Together with the fact that the ##\vec v_i## are orthonormal, yes.
 
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  • #7
It makes sense now, thank you!
 
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