#$&* course MTH 277 09/14/20119:35 am If your solution to stated problem does not match the given solution, you should self-critique per instructions at
.............................................
Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: Let u = <-4,3> and v = <2,-1/2>. Find scalars s and t so that s * <0,3> + tu = v. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: s * <0,3> + t<-4,3> = <2,-1/2> assume s = 1 t<-4,3> = <2, -3 ½ > -4/-2 = 2 3/x = -3 ½ *light bulb * assume t = -2 s<0,3> = <0, 3> s = 1 t = -2 s = 1 confidence rating #$&*:232; 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: The equation s * <0,3> + tu = v becomes s * <0, 3> + t * <-4, 3 > = < 2, -1 / 2 > or <0, 3s > + <-4 t, 3 t > = <2, -1/2 >. and finally <0 - 4 t, 3 s + 3 t > = < 2, -1/2 >. Since the two vectors are equal if an only if their two components are equal, this is equivalent to the two simultaneous equations -4 t = 2 3 s + 3 t = -1/2. The solution of these equations is t = -1/2, s = 1/3. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I needed to combine the problem one step further, it never occurred to me to add the two different variables together before solving them. ------------------------------------------------ Self-critique rating:2 ********************************************* Question: Let u = 4i - 3j, v = -3i + 4j , and w = 6i - 3j. Write the expression ||u|| ||v|| w in standard form. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: ||4i - 3j|| ||3i + 4j|| (6i -3j) sqrt(16i^2 + 9j^2) sqrt(9i^2 + 16j^2) (6i -3j) confidence rating #$&*:232; 1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: || u || = sqrt( 4^2 + 3^2) = 5 and || v || = sqrt(3^2 + 4^2) = 5 so that || u || || v || w = 5 * 5 * w = 25 * (6 i - 3 j ) = 150 i - 75 j. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I have really got to stop making problems more complicated than they really are . . . all I had to do was deal find the magnitude as if the variables weren’t there. ------------------------------------------------ Self-critique rating:2 ********************************************* Question: Let u = 4i + j, v = 4i + 3j, w = -i + 2j. Find a vector of length 3 with the same direction as u - 2v + 2w. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Direction = u/||u|| Direction = (4i + j) - 2(4i + 3j) + 2(-i + 2j) [4 - 2(4) + 2(-1)]i + [1 -2(3) +2(2)]j [-6]i + [-4]j direction = -6i - 4j -6i - 4j = vector/3 -18i - 12j = vector confidence rating #$&*:232; 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: u - 2v + 2w = -6 i - j so || u - 2 v + 2 w || = sqrt(37) and ( u - 2 v + 2 w ) / || u - 2 v + 2 w || = -6 sqrt(37) / 37 i - sqrt(37) / 37 j is a unit vector in the directio of u - 2 v + 2 w . A vector of magnitude 3 in this direction is therefore 3 ( -6 sqrt(37) / 37 * i - sqrt(37) / 37 * j ) = -18 sqrt(37) / 37 i - 3 sqrt(37) / 37 j &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): First mistake, I miscalculated with the j, ending up with a scalar of 4 instead of one. Second mistake, I did not attempt to calculate the using the Pythagorean theorem. ------------------------------------------------ Self-critique rating: 2 ********************************************* Question: Show that the vector v = cos(theta)i + sin(theta)j is a unit vector for any angle theta. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Cos^2x + sin^2x = 1 ||v|| = cos(theta)^2i + sin(theta)^2j ||v|| = 1 any vector found will be a unit vector confidence rating #$&*:232; 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: || v || = sqrt( cos^2(theta) + sin^2(theta) ) = sqrt(1) = 1. A vector of magnitude 1 is a unit vector. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique rating:OK " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!