# Find the slope of the vector that is perpendicular to the...

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Assessment of Learning: Teacher-marked Lesson 20: Intersections of lines and planes in 3-space Properties of vectors You are near the end of this course. This is your final Assessment of Learning , which is used to evaluate your work based on established criteria and to assign a mark. Your teacher will provide you with feedback and a mark. This Assessment of Learning is worth 24% of your final mark for the course. There are four Tasks in this Assessment of Learning . Task 1: Knowledge and Understanding questions 1. Determine the angle between each of the following pairs of vectors. (10 marks: 5 marks each) a) u = ( 2, 4 ) v =( 1, 3 ) First, find the dot product of u and v using the Cartesian formula: u∙ v = ( 2, 4 ) ( 1, 3 ) ¿ ( 2 ) ( 1 ) +(− 4 )(− 3 ) ¿ 2 + 12 ¿ 14 Then find the magnitude of each of the vectors: | u | = ( 2 ) 2 + ( 4 ) 2 ¿ 20 | v | = ( 1 ) 2 + ( 3 ) 2 ¿ 10 Substitute these values into the formula | u | v ¿ cos ( θ )= u∙ v ¿ ( θ ) = u∙ v | u | v ¿ cos ¿
cos θ = 14 ( 20 ) ( 10 ) cos θ = 14 200 cos θ = 0.9899 θ = cos 1 0.9899 θ = 8.1 ° = 8 ° The angle between u and v is 8 ° b) p = ( 1,4,5 ) q =( 3, 1,3 ) First, find the dot product of p and q using the Cartesian formula: p∙ q = ( 1,4,5 ) ( 3, 1,3 ) ¿ ( 1 ) ( 3 ) + ( 4 ) ( 1 ) +( 5 )( 3 ) ¿ 3 4 + 15 ¿ 8 Then find the magnitude of each of the vectors: | p | = ( 1 ) 2 + ( 4 ) 2 + ( 5 ) 2 ¿ 1 + 16 + 25 ¿ 42 | q | = ( 3 ) 2 + ( 1 ) 2 + ( 3 ) 2 ¿ 9 + 1 + 9
¿ 19 Substitute these values into the formula for cos θ : ( θ ) = p∙ q | p | q ¿ cos ¿ cos θ = 8 ( 42 )( 19 ) cos θ = 8 ( 42 )( 19 ) cos θ = 8 798 cos θ = 0.2832 θ = cos 1 0.2832 θ = 73.5 ° = 74 ° The angle between p and q is 74 ° 2. Find the slope of the vector that is perpendicular to the scalar equation 6 x 3 y + 2 = 0. (2 marks) Let u represent the vector that is perpendicular to the scalar equation. Ax + By + C = 0 u =( A ,B ) u =( 6, 3 )
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3. Write an alternate vector equation for the following line. Change both the point and the direction vector: w = ( 4, 1,3 ) + t ( 2,1,7 ) (3 marks) : ) : .
4. Determine whether the angle between each of the following pairs of vectors is acute, obtuse, or neither. (4 marks: 2 marks each) )