Vector calculus position vector

  • MHB
  • Thread starter Chris L T521
  • Start date
In summary, a position vector is a vector that represents the location of a point in three-dimensional space. It is typically represented using the variable <i>r</i> with components <i>x</i>, <i>y</i>, and <i>z</i>. It is an essential tool in vector calculus as it allows for calculations of displacement, velocity, and acceleration. It can also be used to represent curved paths using parametric equations.
  • #1
Chris L T521
Gold Member
MHB
915
0
Thanks again to those who participated in last week's POTW! Here's this week's problem!

-----

Problem: If $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are constant vectors, $\mathbf{r}$ is the position vector $\langle x,y,z\rangle$ and $E$ is given by the inequalities $0\leq \mathbf{a}\cdot\mathbf{r} \leq \alpha$, $0\leq \mathbf{b}\cdot\mathbf{r} \leq \beta$, $0\leq \mathbf{c}\cdot\mathbf{r} \leq \gamma$, show that
\[\iiint\limits_E (\mathbf{a}\cdot\mathbf{r}) (\mathbf{b}\cdot\mathbf{r}) (\mathbf{c}\cdot\mathbf{r}) \,dV = \frac{(\alpha \beta \gamma)^2}{8|\mathbf{a}\cdot(\mathbf{b} \times\mathbf{c})|}\]

-----

 
Physics news on Phys.org
  • #2
No one answered this week's question. Since the solution was such a pain (sorry about not working it out prior to posting the problem), I've typed up this week's solution as a pdf [since I don't think it would be able to fit nicely in one post].

You can find the solution in my public dropbox folder by https://dl.dropboxusercontent.com/u/25818055/MHB_POTW_University_Wk75.pdf.
 

FAQ: Vector calculus position vector

What is a position vector in vector calculus?

A position vector is a mathematical representation of the location of a point in three-dimensional space. It is a vector that starts from the origin and ends at the point, indicating the direction and magnitude of the displacement from the origin to the point.

How is the position vector represented mathematically?

In vector calculus, the position vector is typically represented using the variable r, with its components x, y, and z. It can be written as r = xi + yj + zk, where i, j, and k are unit vectors in the x, y, and z directions respectively.

What is the significance of the position vector in vector calculus?

The position vector is an essential tool in vector calculus as it allows us to describe the location of a point in three-dimensional space and perform operations such as differentiation and integration. It is also used to define other important quantities such as displacement, velocity, and acceleration.

How is the position vector used in vector calculus to calculate displacement?

The displacement from point A to point B can be calculated using the position vectors of the two points, rA and rB. The displacement vector, d, is given by d = rB - rA. This represents the change in position from A to B.

Can the position vector be used to represent curved paths?

Yes, the position vector can be used to represent curved paths in three-dimensional space. This is known as a parametric curve, where the position vector is a function of time, r(t). By varying the value of t, we can trace out different points on the curve, allowing us to represent complex paths in space.

Similar threads

MHB Can You Crack This Vector Identity Challenge?
Replies
2
Views
2K
MHB Problem of the Week #129 - September 15th, 2014
Replies
1
Views
1K
A Virial theorem and translational invariance
Replies
2
Views
1K
MHB Problem of the Week #48 - February 25th, 2013
Replies
1
Views
1K
MHB Is the Inner Product of Matrices Preserved under Vector Multiplication?
Replies
1
Views
1K
I Inner product of two tensors
Replies
6
Views
407
MHB What are the proofs for various equations related to Maxwell's equations?
Replies
1
Views
2K
MHB F convex iff Hessian matrix positive semidefinite
Replies
24
Views
3K
I How to Find the Berry Phase in Slowly Varying Systems
Replies
2
Views
910
MHB Is the Divergence of the Cross Product of Two Gradients Always Zero?
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top