query 105

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@& V(t) is the velocity function, which is the time derivative of the position function:

V(t) = R ' (t).

T(t) is the unit tangent vector. The velocity V(t) is tangent to the curve defined by position vector R(t), so if we divide V(t) by its magnitude we get a unit vector tangent to the curve. So

T(t) = V(t) / || V(t) || = R ' (t) / || R ' (t) ||.

N(t) is the unit normal vector, perpendicular to the tangent. The derivative of a unit vector is perpendicular to that vector. T(t) is a unit vector tangent to the curve, so T ' (t) is perpendicular to the tangent, and is therefore normal to the curve. Thus

N(t) = T ' (t) / || T ' (t) ||

is a unit vector normal to the curve.

A(t) = V ' (t) = R '' ( t ) is the acceleration vector, the second derivative of the position vector.

A(t) lies in the plane of T(t) and N(t), and components parallel to each of these vectors. ( A(t) is therefore a linear combination of T(t) and N(t) ).

The component of a given vector in the direction of another vector is its projection onto that vector

(proj_`w(`v) is the projection of `v onto `w, and has magnitude v cos(theta), where theta is the angle between `v and `w; thus proj_`w(`v) = v cos(theta) * `w / || w ||, where `w / || `w || is the unit vector in the direction of `w; since v cos(theta) = `v dot `w / || `w ||, the projection is `v dot `w * `w / || `w ||^2. If `w is a unit vector then || `w || = 1 and the projection is `v dot `w * `w).

Thus the component of A(t) in the direction of the unit tangent is A_T = A(t) dot T(t), while the component of A(t) in the direction of the unit normal is A_N = A(t) dot N(t). A(t) = A_T * T(t) + A_N * N(t) (a linear combination of the unit tangent and unit normal vectors with constant A_T and A_N).

. *@

@& The above summary confuses, to an extent, vector and scalar projections. The component of one vector in the direction of another can be specified by the scalar projection

proj_`w (`v) = `v dot `w / || `w ||

or the vector projection

`proj_`w (`v) = `v dot `w / || `w || * `w / || `w ||.

The former is just the number that tells you what to multiply by the unit vector in the direction of `w to get the vector component.

The sum

`proj_`T ( `A) + `proj_`N ( `A)

of the vector components in the directions of the unit tangent and unit normal vectors (which we remember are perpendicular) is equal to the vector `A.*@

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@& `B = `T X `N is a unit vector, since `T and `N are perpendicular unit vectors.

`T X `N is perpendicular to `T and `N, since the cross product of two vectors is perpendicular to both.

So `T, `N and `B comprise a 3-dimensional coordinate system, which for a typical curve continuously changes its orientation in space, curving and twisting along with the motion described by `R(t).

If the motion stays in one plane (which will be the plane defined by `T and `N), then `B, being perpendicular to `T and `N, is perpencidular to that plane and hence constant.

If the motion changes planes, the rate of change of `B is a measure of how quickly the plane of motion is changing.

Imagine riding a curving, twisting roller coaster. At any instant you are moving in some direction (if you're looking straight ahead then you're looking in that direction), and being accelerated toward the center of some circle (the osculating circle) with acceleration equal to v^2 / r, where r is the radius of that circle (the reciprocal of the curvature). You are temporarily moving in the plane of that circle, and `B is normal to that plane. If you took three points on the track, one the point at which you are at the instant, one at a point you were a very short time earlier, and another the point at which you will be a very short time later, these three points define a plane which is very close to the plane of the circle. The next three points will typically define a slightly different plane. The `B vector will hence change, and if you divide the change in that vector by the time required to move from one point to the next, you have `B ' .

*@

@& `B ' in the preceding is the rate at which `B changes with respect to time. If you divide that by the rate at which position changes with respect to time, you get the rate at which `B changes with respect to position. That is d`B / ds

Now d`B/ds is formally equal to d `B / dt * dt /ds, which is equal to d`B / dt divided by ds / dt.

s is just arc length. If R(t) = x(t) `i + y(t) `j + z(t) `k, then on a short interval

`ds = sqrt( `dx^2 + `dy^2 + `dz^2) = sqrt( (x ' `dt)^2 + (y ' `dt)^2 + (z ' `dt)^2 ) = sqrt( x ' ^2 + y ' ^2 + z ' ^2) * `dt.

A little thought shows that the result is that

ds / dt = sqrt( x ' ^2 + y ' ^2 + z ' ^2).

So:

If you know `R(t) you can calculate `T(t), `N(t) and `B(t), as well as ds/dt.

So you can calculate all sorts of things that describe the important aspects of motion in 3 dimensions.

In relation to the current question, consider the possibility that dT / ds is in the direction of N, but dN/ds is not necessarily in the direction of T. That and the product rule for the derivative of a cross product can be used to prove the desired result.*@

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@& Check my notes, which pretty much summarize the chapter.

You might have questions, which would be welcome.*@