# 27 5 2 24 c 1 mark 7 2 7 2 2 2 53 unit 5 vectors 7

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∙ ? ⃗ = (2)(−7) + ( −5) ( 2 ) ? ∙ ? ⃗ = −24 c. |? | (1 mark) | ? ⃗ | = (−7, 2) | ? ⃗ | = −7 2 + 2 2 | ? ⃗ | = √ 53
Unit # 5 Vectors 7. Given u = [ 4,4, 4] and v = [ 5,6, 2] find a. 2? ⃗ + 4? ⃗⃗⃗⃗ 2? ⃗ + 4? ⃗⃗⃗⃗ = 2 (−4, 4, −4) + 4(−5, 6, 2) 2? ⃗ + 4? ⃗⃗⃗⃗ = (−8, 8, −8) + (−20, 24, 8) 2? ⃗ + 4? ⃗⃗⃗⃗ = −28, 32, 0 b. ? ∙ ? ? ⃗ ∙ ? = (−4, 4, −4) ∙ (−5,6,2) ? ⃗ ∙ ? = (−4)(−5) + (4)(6) + (−4)(2) ? ⃗ ∙ ? = 20 + 24 − 8 ? ⃗ ∙ ? = 36 c. ? ⃗ ? ? ? ⃗ ? ? = ( −4, 4, −4) ? (−5, 6 , 2) ? ⃗ ? ? = (4)(2) − (−4)(6), (−4)(−5) − (−4)(2), (−4)(6) − (4)(−5)) ? ⃗ ? ? = [ 32, 28, −4] d. |? ⃗ | |? ⃗ | = |[−4, 4, −4]| |? ⃗ | = √−4 2 + 4 2 + (−4 2 ) |? ⃗ | = √48
8. Find u v (1 mark)
9. Find the volume of the parallelepiped defined by the vectors. (2 marks) 7]
Unit # 5 Vectors
Unit # 5 Vectors 10. Given u = [10,4] and v = [12, 13], find the projection of u on v. (2 marks) Note: projection of ? ⃗ ?? ? = ( ? ⃗⃗⃗ ∙ ? ? ? ⃗⃗⃗ )? ? ⃗ ∙ ? = [10, 4] ∙ [12, −13] = (10)(12) + 4 (−130 = 120 − 52 = 68 ? ? = [12, −13] ∙ [12, −13] = (12)(12) + (−13)(−13) = 144 + 169 = 313 ( ? ⃗⃗⃗ ∙ ? ? ? ⃗⃗⃗ ) ? = ( 68 313 ) [12, −13] ( ? ⃗⃗⃗ ∙ ? ? ? ⃗⃗⃗ ) ? = [2.61, −2.83] 11. Given the force, in Newtons, to be [15,12] for an object moving along the vector [7,3] in meters, find the work done. (1 mark) Let w represent work (?) = ? ⃗⃗⃗ ? (?) = [15,12] ∙ [7,3] (?) = (15)(7) + (12)(3) (?) = 141 The volume of the parallelepiped is 398 cubic units. The projection of ? ⃗ ?? ? = = [2.61, −2.83] The work done is 141 J
12. Determine the work done by the force, F, in the direction of the displacement, s. (1 mark) ?
Unit # 5 Vectors
13. Given the points A(3,5) and B( − 3, − 7), find a. The vector equation of the line. (2 marks)
b. The parametric equations of the line. (2 marks) )
c. The symmetric equation of the line. (1 mark) is 𝑖?
Unit # 5 Vectors
Unit # 5 Vectors ? − 3 1 = ? − 5 2 ??