HW9 Sol - Math 2312 Part (1) Practice problems: 1. Find a...

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Math 2312 Part (1) Practice problems: 1. Find a vector with initial point (-1,2) and terminal point (2,-3) Solution: j i ˆ 5 ˆ 3 5 , 3 2 3 ), 1 ( 2 2. 2. (4 pts) Darken only the result of the vector operation on the diagram, clearly indicate direction. (a) Solution: AB AC D C E B A AC AB
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(b) Solution: AD 2 1 D C E B A 3. If 3 , 2 v and j i ˆ ˆ 3 u a) Compute u v u 2 1 Solution: j i ˆ 3 ˆ 2 v and 10 1 9 ) 1 ( 3 2 2 u so,
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j i j i j i j i j i j i ˆ 2 1 10 6 ˆ 2 3 10 4 ˆ 2 1 10 3 ˆ 2 3 10 2 ˆ 2 1 ˆ 2 3 ˆ 10 3 ˆ 10 2 ) ˆ ˆ 3 ( 2 1 ) ˆ 3 ˆ 2 ( 10 2 1 u v u b) Find a vector of length 5 that is in the direction opposite to v . Solution: We first need a unit vector in the direction of v: 13 3 , 13 2 3 , 2 13 1 13 3 , 2 3 ) 2 ( 3 , 2 2 2 v v A unit vector in the opposite direction would be 13 3 , 13 2 13 3 , 13 2 v v and a vector in the opposite direction but with length 5 units would be 13 15 , 13 10 13 3 , 13 2 5 5 v v c) Find the magnitude and the direction angle of v. Solution: 13 v , as we found in part (b), The direction angle is computed from the equation 2 3 tan where the right side is the ratio of the j-component over the i-component of v. We note that the vector v is in the 2nd quadrant – the terminal point is (-2,3). We need to compute the angle between this vector and the positive x-axis.
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If we compute using the arctan function on a calculator, we find -56.31 2 3 arctan 2 3 tan However, we note that the range of arctan is 90 90 (the angles can only be in the 1 st or 4 th quadrants) , so we have really found the angle in the diagram below, which is computed under the assumption that the vector is in the 4 th quadrant, i.e., the calculator reads 3/-2 as -3/2, which represents a vector in the 4 th quadrant, 3 , 2 : -56.31 2 3 arctan 2 3 arctan  
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is the same as that between the vector v and the negative x axis: v Directions angle Therefore, to find the desired direction angle, we subtract 56.31 from 180: 180-56.31=123.69 degrees. d) Compute u v u 2 1 Solution: First, we compute the dot product: 9 ) 3 )( 1 ( ) 2 )( 3 ( 3 , 2 1 , 3 v u Then, 2 9 2 1 v u so that j i ˆ 2 9 ˆ 2 27 2 9 , 2 27 1 , 3 2 9 2 9 2 1 u u v u e) Find the angle between the vectors u and v. Round to 2 decimal digits. Solution:
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HW9 Sol - Math 2312 Part (1) Practice problems: 1. Find a...

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