A spherical alien spaceship lands at rest on a Earth surface given by z = 2x^2 −3y^2 at a point A(2,1,5). Find the unit vector of the direction in which the ship will roll (assume the aliens have a faulty ship with no legs or breaking devices to opose the fall). Sketch the surface and identify the landing position.

Guest Mar 31, 2015

#8**+10 **

I've found the following on-line 3d graphing facility that seems to be pretty good (though I had some difficulty controlling the orientation with the rotation tool!). It can be found at: http://web.monroecc.edu/manila/webfiles/calcNSF/JavaCode/CalcPlot3D.htm

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Alan
Apr 5, 2015

#1**+10 **

Small changes $$\delta x$$ and $$\delta y$$ in the x,y co-ordinates will bring about a small change $$\delta z$$ in the z co-ordinate given by

$$\delta z \approx \frac{\partial z}{\partial x}\delta x+ \frac{\partial z}{\partial y}\delta y.$$

So, $$\delta z \approx 4x \delta x - 6y \delta y.$$

We need $$\delta z$$ to be negative, in which case we need $$\delta x$$ to be negative and $$\delta y$$ to be positive. So, the move needs to be into the second quadrant of the xy plane.

If then we move a small distance $$h$$ at an angle $$\theta$$ to the negative x-axis, $$\delta x = -h\cos \theta,$$ and $$\delta y = h \sin \theta,$$ so $$\delta z \approx -h(8\cos\theta+6\sin\theta).$$

This will be a maximum when $$\tan \theta = 3/4$$ from which it follows that a vector in the required direction will be $$-4\hat{\imath}+3\hat{\jmath}$$ and a unit vector will be that divided by 5.

Bertie
Apr 1, 2015

#2**+10 **

That looks impressive Bertie but where is my picture

I did graph it and I got a very interesting landscape but I didn't know what to do with A.

I didn't find a 3D grapher that would plot both the graph and the point as well.

I'd really like to see them together. Maybe I could get a 'feel' for the question that way.

That is hopeful thinking that is :)

Actually, this graph is not too bad:)

Here is a graph of the terrain around landing spot. :)

Melody
Apr 2, 2015

#5**+10 **

Here's a picture Melody:

The copper colour is the z = 2x^{2} - 3y^{2} surface. The red blob is the point A = (2, 1, 5). The blue mesh is the plane tangential to the surface at A (z = 8x - 6y - 5).

The black line is the unit length vector lying in the tangential plane in the direction of steepest descent.

I took Bertie's solution for the vector in the horizontal plane, found where the end of this would meet the tangential plane, scaled the resulting vector from A to this point to be of unit length and plotted the black line to represent the unit vector in the tangential plane.

The black line goes from (2, 1, 5) to (1.92, 1.06, 4.01) (approximately).

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Alan
Apr 3, 2015

#7**+5 **

I used the graphing facilities in MATLAB. Unfortunately, this is a commercial program, not free open-source.

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Alan
Apr 5, 2015

#8**+10 **

Best Answer

I've found the following on-line 3d graphing facility that seems to be pretty good (though I had some difficulty controlling the orientation with the rotation tool!). It can be found at: http://web.monroecc.edu/manila/webfiles/calcNSF/JavaCode/CalcPlot3D.htm

.

Alan
Apr 5, 2015