Unit 7 Calculus Final.docx - Unit 7 Final Assignment Task 1...

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Unit 7: Final Assignment Task 1: Knowledge and Understanding questions 1. Determine the angle between each of the following pairs of vectors. θ θ
Find magnitude ¿ ( 1 ) 2 + ( 4 ) 2 + ( 5 ) 2 ¿ 42 ¿ ( 3 ) 2 + ( 1 ) 2 + ( 3 ) 2 ¿ 19 Substitute into | u || v | coscos θ u v = ( 42 ) ( 19 ) cos cos θ Find dot product u v = ( 1 ) ( 3 ) + ( 4 ) ( 1 ) + ( 5 ) ( 3 ) u v = 8 Substitute into u v = ( 2 5 ) ( 10 ) coscos θ and solve for θ 8 = ( 42 ) ( 19 ) cos cos θ 8 798 = cos cos θ ( 8 798 ) = θ 73.5 ° = θ 2. Find the slope of the vector that is perpendicular to the scalar equation :
Therefore, the slope of the vector that is perpendicular to the scalar equation 6 x 3 y + 2 = 0 is m = 1 2 . 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 ) Add the directional vector into the point on the line to find another point. .
4. Determine whether the angle between each of the following pairs of vectors is acute, obtuse, or neither. θ
u v = ( 3 13 ) ( 17 ) coscos θ Find the dot product u v = ( 10 ) ( 3 ) + ( 4 ) ( 2 ) + ( 1 ) ( 2 ) ¿ 4 0 Substitute into u v = ( 2 5 ) ( 10 ) coscos θ and solve for θ 40 = ( 3 13 )( 17 ) cos cos θ 40 3 221 = coscos θ ( 40 3 221 ) = θ 26.2 ° = θ Therefore, the angle is acute. b. p = ( 0,4 , 3 ) q = ( 7 , 2,1 ) u v = | u || v | cos cos θ Find magnitude ¿ ( 0 ) 2 + ( 4 ) 2 + ( 3 ) 2 ¿ 5 ¿ ( 7 ) 2 + ( 2 ) 2 + ( 1 ) 2 ¿ 3 6 Substitute into | u || v | coscos θ u v =( 15 )( 3 6 ) cos cos θ Find the dot product u v = ( 0 ) ( 7 ) + ( 4 ) ( 2 ) + ( 3 ) ( 1 ) u v =− 11 Substitute into u v =( 15 )( 3 6 ) cos cos θ and solve for θ 11 = ( 15 ) ( 3 6 ) cos cosθ
11 ( 15 ) ( 3 6 ) = coscos θ ( 11 ( 15 ) ( 3 6 ) ) = θ 95.7 ° = θ Therefore, the angle is obtuse. 5. Given the vector equation of a line in 2-space, ( x, y ) =( 3,2 )+ t ( 2,4 ) , write a scalar equation for the line.

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