# Tagged Questions

In $\Bbb R^3$, the cross product of two vectors $v$ and $w$ produces a vector $v \times w$ perpendicular to both. This tag is not meant for products in other mathematical contexts, such as products of groups (such as the [tag:direct-product]), sets (the Cartesian product), graphs, and so on.

89 views

### Can the cross product of two non-invertible matrices be invertible?

To put it better, if A and B are non-invertible matrices (for whatever reason), can the matrix AB be invertible? Just used to help understand a Linear Transformation assignment question, don't ...
95 views

### How can we determine if two vectors are parallel?

What are the minimal number of products like dot cross that can give us information if two vectors are parallel ? What can we say if V*W = 1 assuming V and W are not unit vectors.
52 views

### How to calculate one of the vectors that generate a given cross-product?

Given the vector: $$\vec b=(-0.361728, 0.116631, 0.924960)$$ and it's cross-product: $$\vec a \times \vec b=(-0.877913, 0.291252, -0.380054)$$ How do I calculate $\vec a$ ? It's been a while since I'...
54 views

### Lack of associativity of cross product vs associativity of the exterior product

Can someone remind me in a nutshell why the associativity of the exterior product fails to transfer to the cross product? (It's been over a decade since I had to deal with the former back in school.) ...
72 views

### Higher dimensional cross product

I know that cross products do not exist in 4, 5 or 6 dimensions, but do in 7 dimensions. So I was wondering if this was because cross products can be considered the imaginary part of $2^n - ion$ ...
27 views

48 views

### A high-level reason that $u \cdot (v \times w) = (u \times v) \cdot w$?

I can do the algebra to show that for $u, v, w \in \mathbb{R}^3$, this identity is true: $$u \cdot (v \times w) = (u \times v) \cdot w$$ But is there a more high-level reason? I didn't expect the ...
53 views

### Vector calculus problem

I have to solve this: $$[(\nabla \times \nabla)\cdot \nabla](x^2 + y^2 + z^2)$$ But I am really drowning in the sand.. Can anybody help me please?
41 views

### Trying to show $|\overrightarrow{a}\times\overrightarrow{b}|^2=|\overrightarrow{a}|^2|\overrightarrow{b}|^2-(\overrightarrow{a}⋅\overrightarrow{b})^2$

If $\overrightarrow{a} = \langle a_1, a_2, a_3 \rangle$ and $\overrightarrow{b} = \langle b_1, b_2, b_3 \rangle$, then the cross product of $\overrightarrow{a}$ and $\overrightarrow{b}$ is the vector ...
30 views

43 views

### Why is the cross product a x b dependent on size of vector a?

If I consider a x b = c, as a system where the vector b is rotating about an axis defined by vector a, and vector c shows the linear direction which vector b moves as it is rotating. The faster the ...
34 views

### Can you relate the cross product of two vectors to rotational motion.

Say you have a cross product a x b = c. Can you intepret this as the vector b spinning about the vector a. If the angle between them is close to zero then vector b is close to zero and is spinning ...
I don't think I ever understood the rationale behind this. I get that the dot product $\mathbf{a} \cdot \mathbf{b} =\lVert \mathbf{a}\rVert \cdot\lVert \mathbf{b}\rVert \cos\theta$ is derived from ...