# 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 solve proportions involving vector cross products?

I have the following proportion $\vec{JV} \times \vec{F_v} = \vec{JM} \times \vec{F_m}$ and all members are known except the magnitude of the vector $\vec F_m$, like described by another question here ...
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### Is the angle between a and b is equal to the angle between b and a?

This was a question in an exam: Calcualte tan of the angle between a and b if:a = (4,3) and b = (5, -12) There are two answers to this question: Some students devided the dot product of a and b by ...
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### What is the logic/rationale behind the vector cross product?

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 ...
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### Cross product angle formula

Say there are two vectors $A$ and $B$ in $3D$. to get the angle between the cross product of those two vectors, you use $$||A\times B|| = ||A||\;||B||\sin(\theta).$$ right? Is this equation ...
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### Proof of $(A+B) \times (A-B) = -2(A X B)$ [closed]

Proof of $(A+B) \times (A-B) = -2(A \times B)$, where 'A' and 'B' are vectors
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### Show that ∇· (∇ x F) = 0 for any vector field [duplicate]

To solve this question, how do I define any vector field $F$, in order to solve it? I called $F = (ax,by,cz)$, in which case already $\nabla\times F = 0$. How would i go about proving this? Many ...
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### Which sides of a triangle are visible to an observator?

Working o 2-d plane. Supposing that there is a observer standing on the origin (0, 0) looking to the first quadrant. If there is a triangle drawn on the first quadrant, what sides are visible to the ...
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### Cross products and orthogonal complements

I am having trouble with this question about cross products and orthogonality: Let a ∈ R3 \ {0} Show that if y ⊥ a then $\exists$ x {x ∈ R3 : a × x = y} Could anyone explain this to me? Thanks
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### Cross Product in 3D

Hi! I am currently working on some calc2 online homework problems concerning the cross product. I understand how the cross product works, but I am not sure how to apply it to this question. I know ...
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### Determine Cross Product with Left Hand vs Right Hand

If I perceive http://en.wikipedia.org/wiki/Cross_product correctly, then to determine the cross product With a right hand, let: the 1st vector in the cross product = your index finger = in red ...
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### Arbitrary Dot and Cross Products

I am having a bit of trouble with answering these few dot and cross product questions. Suppose that $u · (v × w) =3$. Find, $w · (u × v)$ $v · (u × w)$ $(u × w) · v$ Could some explain their ...
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### Showing the distance between a point $P$ the line determined by a segment $AB$ is $d=\frac{||AP\times AB||}{||AB||}$

Show that in $3$-space the distance $d$ from a point $P$ to the line $L$ through points A and B can be expressed as $$d=\frac{||AP\times AB||}{||AB||} .$$ My diagram of the situation: My next ...
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### Logic behind cross and dot products

Let $A, B, C,$and $D$ be four distinct points in $3-space$. If $AB×CD$ does not equal $0$ and $AC⋅(AB×CD)=0$, explain why the line through $A$ and $B$ must intersect the line through $C$ and $D$. ...
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### Question about cross product of images of linear transformation

I'm reading "Differential Geometry: Curves and Surfaces" of Manfredo Carmo, and this part in the book confuses me(page 166): Suppose that $N: S \rightarrow S^2$ is the Gauss map of regular surface ...
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### How to prove that the cross product of two vectors is a linear transformation?

I'm stuck, could I please have a hint?
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### Origin of the dot and cross product?

Most questions usually just relate to what these can be used for, that's fairly obvious to me since I've been programming 3D games/simulations for a while, but I've never really understood the inner ...
I do not know how to prove this, can anybody help me out with that? Consider five vectors: $\vec{a},\vec{b},\vec{c}, \vec{p}, \vec{q} \in \mathbb{R}^3$ then: (\vec{p}\cdot\vec{q})(\vec{a}\cdot(\...
Say you had $(a_x,a_y,a_z)\times(b_x,b_y,b_z)$, you would set up a matrix like the following: And the resulting would be your Cross Product or the coordinates of an orthogonal vector. My question ...