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Math 206A Quiz 03 - Line Intersection and Vector-Valued Function, Exercises of Calculus

University of AllahabadCalculus

A math quiz problem from a university course, math 206a. The problem involves finding the point of intersection between two lines, the vector that is perpendicular to the plane determined by the lines, and the equation of that plane. Additionally, the problem asks for a sketch of a vector-valued function and the values of t for which the path crosses the y-axis.

Typology: Exercises

2012/2013

Uploaded on 03/21/2013

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Math 206A Quiz 03 page 1 10/01/2010 Name
1. Consider the two lines parameterized by โˆ’โ†’
`1(t) = โˆ’โˆ’โˆ’โˆ’โ†’
(3,4,5) + t
โˆ’โˆ’โˆ’โˆ’โ†’
(1,2,3) and โˆ’โ†’
`2(s) = โˆ’โˆ’โˆ’โˆ’โˆ’โˆ’โ†’
(โˆ’4,2,8) + s
โˆ’โˆ’โˆ’โˆ’โ†’
(3,2,1), respectively.
1a. What point Pis on the first line (โˆ’โ†’
`1), when t= 2?
1b. It turns out this same point Pis on the second line. Find the value of sfor which โˆ’โ†’
P=โˆ’โ†’
`2(s).
1c. Since Pis on both lines, these lines determine a plane. Find a vector nwhich is perpendicular to that plane.
1d. Find the equation of that plane, in the form ax +by +cz =d.
2A. Use your calculator do make a good sketch of the path given by the vector-valued function
โˆ’โ†’
f(T) =
โˆ’โˆ’โˆ’โˆ’โˆ’โˆ’โˆ’โˆ’โˆ’โˆ’โˆ’โˆ’โˆ’โˆ’โ†’
(T(T2โˆ’3), T 2โˆ’4) = T(T2โˆ’3)i+ (T2โˆ’4)jon the domain Tโˆˆ[โˆ’2.25,2.25].
2B. For what values of Tdoes this path cross the y-axis?

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Math 206A Quiz 03 page 1 10/01/2010 Name

  1. Consider the two lines parameterized by

` 1 (t) =

(3, 4 , 5) + t

(1, 2 , 3) and

` 2 (s) =

(โˆ’ 4 , 2 , 8) + s

(3, 2 , 1), respectively. 1a. What point P is on the first line (

` 1 ), when t = 2? 1b. It turns out this same point P is on the second line. Find the value of s for which

P =

` 2 (s). 1c. Since P is on both lines, these lines determine a plane. Find a vector n which is perpendicular to that plane. 1d. Find the equation of that plane, in the form ax + by + cz = d. 2A. Use your calculator do make a good sketch of the path given by the vector-valued function โˆ’โ†’ f (T ) =

(T (T 2 โˆ’ 3), T 2 โˆ’ 4) = T (T 2 โˆ’ 3)i + (T 2 โˆ’ 4)j on the domain T โˆˆ [โˆ’ 2. 25 , 2 .25]. 2B. For what values of T does this path cross the y-axis?