Understanding the Dot Product and Cross Product in Vector Calculations

[jolly_math]

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

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.

 

[Orodruin]

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.

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 $\vec v_1\times\vec v_3$.

 

[jolly_math]

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?

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.

 

[Orodruin]

but what does (v1 · v2)v1 equal?

What is $\vec v_1\cdot \vec v_2$ if all $\vec v_i$ are orthonormal?

Why would v→1×v→3 = −v→2?

Because you know that the $\vec v_i$ form an orthonormal basis and that $\vec v_1\times\vec v_2=\vec v_3$.

 

[jolly_math]

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.

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 $\vec v_1\times\vec v_3 = -\vec v_2$?

 

[Orodruin]

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.

 

[jolly_math]

It makes sense now, thank you!