If your solution to stated problem does not match the given solution, you should self-critique per instructions at

Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

Question:

Section 10.2

The velocity vector corresponding to position vector `R(t) = x(t) `i + y(t) `j + z(t) `k is the derivative `v(t) = `R ' (t) = x ' (t) `i + y ' (t) `j + z ' (t) `k, and the acceleration vector is `a(t) = `v ' (t) = `R '' (t) = x '' (t) `i + y ''(t) `j + z '' (t) `k.

The unit tangent vector is the vector function `T(t), equal at every instant to the unit vector in the direction of the velocity `v(t).

The acceleration vector has components `a_T(t) in the direction of the unit tangent vector, and `a_N(t) = `a(t) - `a_T(t) in the direction perpendicular to the unit tangent vector.

The unit normal vector is the unit vector in the direction of `a_N(t), and is perpendicular to the unit tangent vector.

The direction of the derivative `T ' (t) of the unit tangent vector is the same as that of the unit normal vector.

The unit binormal vector `B(t) is the cross product of the unit tangent and unit normal vectors.

Note that ` in front of a symbol indicates that the symbol is a vector. The only exception: `d means 'Delta'. I will eventually search-replace the document to convert the notation to boldface.

If `R(t) = sin(t) `i + cos(`t) j + t `k then:

`q001. What are the associated velocity and acceleration vectors?

Your solution:

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Given Solution:

`v(t) = `R ' (t) = cos(t) `i - sin(t) `j + `k

`a(t) = `v ' (t) = `r '' (t) = -sin(t) `i - cos(t) `j

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Question: `q002. What is the function describing the unit tangent vector?

Your solution:

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Given Solution:

Divide `v(t) by || `v(t) || and simplify

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Question: `q003. What is the component of the acceleration vector in the direction of the unit tangent vector?

Your solution:

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Given Solution:

The component is denoted `a_T (t) . The desired component is the projection of `a(t) on `T(t).

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Question: `q004. What is the component of the acceleration vector in the direction perpendicular to the unit tangent vector?

Your solution:

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Given Solution:

Subtract the component `a_T(t) from `a(t).

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Question: `q005. What is the normal component of the acceleration?

Your solution:

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Given Solution:

This is the component perpendicular to the unit tangent vector.

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Question: `q006. Show that the normal component of the acceleration is perpendicular to the tangential component.

Your solution:

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Given Solution:

Two vectors are perpendicular if their dot product is zero.

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Question: `q007. Show that the direction of the derivative of the unit tangent vector is the same as that of the unit normal vector.

Your solution:

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Given Solution:

Two vectors are parallel if the cosine of the angle between them is zero.

How therefore can to test to see if the vectors are parallel?

What further test allows us to determine if they are in the same direction, vs. in the opposite directions.

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Question: `q008. Find the unit normal vector.

Your solution:

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Given Solution:

You have at least one vector in the normal direction (in fact in the preceding questions you have found two). Use either to find the unit normal.

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Question: `q009. Find the unit binormal vector.

Your solution:

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Given Solution:

You should have the unit normal and unit tangent. Use them to easily find the unit binormal. How do you know that your result is a unit vector?

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Question: `q009. What difference would it make in the above results if the function was `R(t) = sin(t^2) `i + cos(t^2) `j + t `k?

Your solution:

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Given Solution:

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Question: `q010. What difference would it make in the above results if the function was `R(t) = sin(t^2) `i + cos(t^2) `j + t^2 `k?

Your solution:

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Given Solution:

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