# 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.

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### How to plot a contour in the plane defined by a vector cross-product

I have some contours of a structure in 3D that I would like to plot in 2D, but how do I get from 3D to 2D? In other words, I would like to plot the contour in the plane defined by the cross-product of ...
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### Vector cross products proof

Explain why $U \times ( V \times W )$ must be a vector that satisfies the equation $X = sV + tW$, where $U$, $V$, and $W$ are vectors in $\mathbb R^3$.
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### Multiplying Fractions Help

How could I find each product or quotient for this problem? I got $$\frac{3q}{2r^2}$$ as an answer The question is $$\frac{(4r)^2}{q} \times \frac{(3q)^2}{(8r)^4}$$ . So is my answer correct ?
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### Geometric proof of the Cross Product magnitude

Most proofs of the magnitude of the cross product are algebraic in nature, I find I learn best visually / geometrically. Is there a breakdown of the proof of the magnitude of the cross product using ...
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### Why is cross product not commutative?

Why, conceptually, is the cross product not commutative? Obviously I could simply take a look at the formula for computing cross product from vector components to prove this, but I'm interested in why ...
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### $\vec{a}\times(\vec{a}\times\vec{R})-\vec{b}\times(\vec{b}\times\vec{R})$

I have $\vec{a}\times(\vec{a}\times\vec{R})-\vec{b}\times(\vec{b}\times\vec{R})$, my textbook says that this equals $((\vec{a}\times\vec{a})-(\vec{b}\times\vec{b}))\times\vec{R}=-(a^2-b^2)\vec{R}$. I ...
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### rotation matrix and vector - understand step calculation

I have an extremely equation, but I just don't understand which step they made to get to the last line. ${\bf W}$ and ${\bf V}$ are all 3d vectors. A is a rotation matrix. How did they get that ...
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### Dot and Cross Product Proof: $u \times (v \times w) = ( u \cdot w)v - (u \cdot v)w$

How do you prove that: $u \times (v \times w) = ( u \cdot w)v - (u \cdot v)w$ ? The textbook says as a hint to "first do it for $u=i,j$ and $k$; then write $u-xi+yj+zk$ but I am not sure what that ...
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### Cross Product and Scalar Product proof

Hi i'm trying to prove the following equality where P,Q and R are any 3D vectors: PxQxR = (P.R)Q - (Q.R)P I find it easier by proving that the x coordinate of the left side is equal do the right ...
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### Proof of identity: cross product of three vectors

A book I'm reading contains the following (paraphrased) $$(a \times b) \times c = (a \cdot c)b - (b \cdot c)a$$ This is supposed to follow from: (a \times ...
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### Manipulation of three vectors in $\Bbb R^3$

Please excuse the non-specific title, this is a rather long problem. So on our last exam in multivariable calculus, our professor gave us a very lengthy vector manipulation problem as a bonus. Seeing ...
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### Intuitively, what makes two vector parallel

I have heard explanations such as it is when the cross product equals zero or that it is when one is a scalars multiple of the other but I have not seen an intuitive explanation. Is it when two ...