#$&* course Mth 277 10/02/2011 @ 3:33p.m. I have had this done for several days but have been neglecting to take the time to submit this and our other assignments due last Wednesday, so I'm sorry for the pretty large number of submissions going out today. If your solution to stated problem does not match the given solution, you should self-critique per instructions at
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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 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating:
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Given Solution: Divide `v(t) by || `v(t) || and simplify &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q003. What is the component of the acceleration vector in the direction of the unit tangent vector? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: acceleration vector projected on a tangent unit vector.
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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). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: moderately ********************************************* Question: `q004. What is the component of the acceleration vector in the direction perpendicular to the unit tangent vector? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: ‘a(t) - ‘a_T(t)
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Given Solution: Subtract the component `a_T(t) from `a(t). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q005. What is the normal component of the acceleration? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The component perpendicular to ‘T(t)
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Given Solution: This is the component perpendicular to the unit tangent vector. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q006. Show that the normal component of the acceleration is perpendicular to the tangential component. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If a_’N(t) dot ‘T(t) = 0, then they are perpendicular confidence rating #$&*:y ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Two vectors are perpendicular if their dot product is zero. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q007. Show that the direction of the derivative of the unit tangent vector is the same as that of the unit normal vector. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: cos(’theta) = 0 means they are in the same direction (parallel or the same vector entirely) confidence rating #$&*:y ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Two vectors are parallel if the cosine of the angle between them is zero. How therefore can to test to see if the vectors are parallel? What further test allows us to determine if they are in the same direction, vs. in the opposite directions. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Won’t they be multiples with different signs if they are parallel in different directions or is this going to just make the vectors different entirely? ------------------------------------------------ Self-critique rating: ********************************************* Question: `q008. Find the unit normal vector. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To make any vector a unit vector, we simply divide it by its magnitude. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q009. Find the unit binormal vector. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: We first f ind a unit vector in the same direction as the normal acceleration then cross it with ‘T(t) . confidence rating #$&*: very confident ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: You should have the unit normal and unit tangent. Use them to easily find the unit binormal. How do you know that your result is a unit vector? &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I essentially approached this question by using your equations at the top, but I want to divide by the magnitude of some vector somewhere to find the “unit” part of the binormal vector…but this doesn’t seem to be the case here. ------------------------------------------------ Self-critique rating: ********************************************* Question: `q009. What difference would it make in the above results if the function was `R(t) = sin(t^2) `i + cos(t^2) `j + t `k? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: When we took the derivatives of this function to find velocity and acceleration, the derivate of the “inside” would be 2t instead of just 1 and our 2nd derivative used to find acceleration would have the derivate of our inside times the derivative of 2t*-cos(t^2)i instead of just -cos(t)i. It appears that the acceleration and velocity are just probably going to just be quite a bit larger in this t^2 equation. confidence rating #$&*:y ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* 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? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: k wouldn’t disappear by the time we took the 2nd derivative, so we would accelerate in the k direction. confidence rating #$&*:y ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!
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*@ " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!Be sure to include the entire document, including my notes.
If my notes indicate that revision is optional, use your own judgement as to whether a revision will benefit you.
*@ #*&!