#$&* course Mth 277 9/14 12am If your solution to stated problem does not match the given solution, you should self-critique per instructions at
.............................................
Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: Find the standard form equation of the sphere with center (-1,2,4) and radius 2. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: (x+1)^2+(y-2)^2+(z-4)^2=4 confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: A point (x, y, z) is on the given sphere if its distance from (-1, 2, 4) is 2, so that sqrt( (x - (-1))^2 + (y - 2)^2 + (z - 4)^2 ) = 2 and (x + 1)^2 + (y - 2)^2 + (z - 4)^2 = 4. This is the equation of the sphere in one form. Expanding the squares we obtain x^2 + 2 x + 1 + y^2 - 4 y + 4 + z^2 - 8 x + 16 = 4 which we rearrange to the standard form x^2 + 2 x + y^2 - 4 y + z^2 - 8 z + 13 = 0. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique rating:OK ********************************************* Question: Find the center and radius of the sphere with equation x^2 + y^2 + z^2 - 2x - 6y + 12z - 17 = 0. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: (1,3,-6) r=3sqrt(7) confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: Completing the squares we obtain (x^2 - 2 x + 1 - 1) + (y^2 - 6 y + 9 - 9) + (z^2 + 12 z + 36 - 36) = 17 which can be written as (x - 1)^2 - 1 + (y - 3)^2 - 9 + (z + 6)^2 - 36 = 17 and finally as (x - 1)^2 + (y - 3)^2 + (z + 6)^2 = 63 This sphere is centered at (1, 3, -6) and has radius sqrt(63) = 3 sqrt(7). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):Ok ------------------------------------------------ Self-critique rating:OK ********************************************* Question: Find the standard representation and length of PQ when P = (-3,1,4) and Q = (2,-4,-3). YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: PQ=5i-5j-7k length=sqrt(99) confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: PQ = (2 - (-3) ) i + (-4 - 1) j + (-3 - 4) k = 5 i - 5 j - 7 k. || PQ || = sqrt( 5^2 + 5^2 + 7^2) = sqrt(99). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique rating:OK ********************************************* Question: Find a unit vector in the direction of v = <-1, sqrt(3), 4>. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: =(-1/2sqrt(5))i+(sqrt(15)/10)j+2sqrt(5)/5k confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: || v || = sqrt( 1^2 + sqrt(3) ^ 2 + 4^2 ) = sqrt( 26 ) so a unit vector in the direction of v is v / || v ||= < -1, sqrt(3), 4 > / sqrt(26) = <-sqrt(26) / 26, sqrt(78) / 26, 4 sqrt(26) / 26)> . 4 sqrt(26) / 26 is 2 sqrt(26) / 13. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique rating:OK ********************************************* Question: Sketch and describe the cylindrical surface given by y = cos x. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: A wavy wall confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: In the x-y plane y = cos(x) consists of a sinusoidal function oscillating between the lines y = -1 and y = 1, with period 2 pi radians, and containing the point (0, 1). The surface in 3 dimensions repeats this same curve for every value of z, so that the graph represents a wavy curtain hanging vertically downward, intersecting the xy plane along the sinusoidal curve. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique rating:OK ********************************************* Question: Determine if u = 2i + 3j + -4k is parallel to v = <1,-3/2,2>. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: NO. The components do not even have the same signs confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: Two vectors are parallel if the angle between them is 0 or pi radians (180 degrees), meaning that the cosine of the angle between them is 1 or -1. u dot v = || u || || v || cos(theta) so that cos(theta) = u dot v / (|| u || || v || ) = (2 * 1 + 3 * (-3/2) + (-4 * 2) ) / ( sqrt(2^2 + 3^2 + 4^2) * sqrt( 1^2 + (3/2)^2 + 2^2) ) = (-21/2) / (sqrt( 29) sqrt(29/4). This is not 1 or -1, so the cosine is neither 0 nor pi rad (i.e., 180 deg). The vectors are therefore not parallel. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):Ok I understand how to find it using dot product. ------------------------------------------------ Self-critique rating:2
.............................................
Given Solution: The sides can be represented by the vectors AB = < 4, 1, 4 >, BC = < -2, 3, 0 > and AC = < 2, 4, 4 >. The magnitudes of these vectors are respectively sqrt(33) sqrt(13) sqrt(36). None of the sides are the same length so the triangle is not isosceles. The sum of the squares of the shorter two side is 33 + 13 = 46, which is not equal to the sum of the longest, so the triangle is not a right triangle. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique rating:OK " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!