Multivar Class 110914

The link below is downloadable and/or playable.  A recent version of Windows Media Player appears to run it fine.  QuickTime should run it well.  Let me know.

../../class_notes_current/pendulum_in_changing_water_depth/pendulum_in_changing_water_depth-iPhone.m4v

This doesn't appear to be downloadable to Windows.  If you're running a Mac or IPad give it a try and let me know.  Also might work on a phone.

../../class_notes_current/pendulum_in_changing_water_depth/pendulum_in_changing_water_depth-iPhone-cell.3gp

A line through the point P_0 = (x0, y0, z0) in the direction of the vector v = a i + b j + c k is characterized by the set of points Q = (x, y, z) for which the vector PQ = (x - x0) i + (y - y_0) j + (z - z_0) k is parallel to v.

• Two vectors are parallel if and only if one is a scalar multiple of the other.
• This condition can therefore be expressed as PQ = t * v, for some value of t.
• Writing this out in terms of x0, y0, z0 and t we have

(x - x0) i + (y - y_0) j + (z - z_0) k = t ( a i + b j + c k ).

• Two vectors are equal if and only if each of their components are equal, so our equation is equivalent to the three simultaneous equations

x - x0 = t a

y - y0 = t b

z - z0 = t c.

• We can solve each of these equations for t, obtaining the result t = (x - x0) / a = (y - y0) / b = (z - z0) / c.  Whatever we call t, and whatever value it might have for any given point, the result is that

(x - x0) / a = (y - y0) / b = (z - z0) / c

This is called the symmetric form of the equation of a straight line in xyz space.

• We can also solve our three equations to get equations for x, y and z.  Our equations are

x = x0 + a t

y = y0 + b t

z = z0 + c t.

This is the parametric form of the equation for a straight line in xyz space.

A plane through the point P = (x0, y0, z0) perpendicular to a vector N = a i + b j + c k has the property that the vector PQ = (x - x0) i + (y - y_0) j + (z - z_0) k from P to Q = (x, y, z) is perpendicular to N.  Two vectors are perpendicular if their dot product is zer0, so the condition for Q = (x, y, z) to lie on the plane is

PQ dot N = 0

which we write out as

( (x - x0) i + (y - y_0) j + (z - z_0) k ) dot (a i + b j + c k) = 0.

Writing out the dot product we have

a (x - x0) + b ( y - y0) + c ( z - z0) = 0

This is the most direct form of the equation of our plane.  This equation is connected in a direct and obvious way to the coordinates (x0, y0, z0) of our known point and the normal vector a i + b j + c k.

Using the distributive law we can write this as

a x + b y + c z + (-a x0 - b y0 - c z0) = 0,

or

a x + b y + c z + d = 0.

The latter is the standard form of the equation of a plane in xyz space.

Suggested exercise.  Think through it, do it, or ignore it as you think will best benefit you.  Ignoring it could be beneficial if the time you save is used to understand the text, the qa, or anything else that helps you master these ideas, but if you're in a position to do so it won't take you long to just think through it and get the image in your head.

`q001.  What are the coordinates of the two vectors you estimated along the edges of your fundamental shape (the one you did using the foam chunk)?  You can of course use my coordinates, but you'll relate to this better if you use your own.

What is the cross product of these two vectors?

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What are the coordinates of the point from which your two vectors originated?

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Your answers to the last two questions give you a point (x0, y0, z0) on the plane, and a vector a i + b j + c k which is perpendicular, or normal, to the plane of the top of your foam chunk.  So what is the equation of that plane?

You estimated the coordinates of at least one point where the plane meets the xy plane.  If you plug the coordinates of that point into the equation of the plane, how close is the left-hand side to zero?

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What is the vector from your (x0, y0, z0) point to the opposite corner of the top of the foam chunk?

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What therefore is the equation of the straight line through the two corners of the chunk, in the form (x - x0) i + (y - y0) j + (z - z0) k = t ( a i + b j + c k )?

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What is the equation in symmetric form?

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What is the equation in parametric form?

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Can you use the equation to figure out where that line would meet the xy plane?

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