Vectors #4 - Questions and Answers

Question 1 Vector Dot Product Learning Goal: To understand the rules for computing dot products. Let vectors 𝐴 = ( 2 , 1 , − 4 ) A=(2,1,−4), 𝐵 = ( − 3 , 0 , 1 ) B=(−3,0,1). Angle between the vectors A and B. What is the angle 𝜃 𝐴 𝐵 θ AB ​ between A and B? Express your answer numerically to three significant figures in radians. 𝜃 𝐴 𝐵 = θ AB ​ = radians. Answer 1 𝐴 ⋅ 𝐵 = ∣ 𝐴 ∣ ∣ 𝐵 ∣ cos ⁡ ( 𝜃 ) A⋅B=∣A∣∣B∣cos(θ) Given: 𝐴 ⋅ 𝐵 = − 10 A⋅B=−10 Find ∣ 𝐴 ∣ ∣A∣: ∣ 𝐴 ∣ = 2 2 + 1 2 + ( − 4 ) 2 = 4 + 1 + 16 = 21 . ∣A∣= 2 2 +1 2 +(−4) 2 ​ = 4+1+16 ​ = 21 ​ . Find ∣ 𝐵 ∣ ∣B∣: ∣ 𝐵 ∣ = ( − 3 ) 2 + 0 2 + 1 2 = 9 + 0 + 1 = 10 . ∣B∣= (−3) 2 +0 2 +1 2 ​ = 9+0+1 ​ = 10 ​ . Substitute into the formula: cos ⁡ ( 𝜃 ) = 𝐴 ⋅ 𝐵 ∣ 𝐴 ∣ ∣ 𝐵 ∣ = − 10 21 10 = − 0.69006. cos(θ)= ∣A∣∣B∣ A⋅B ​ = 21 ​ 10 ​ −10 ​ =−0.69006. Find 𝜃 θ: 𝜃 = cos ⁡ − 1 ( − 0.69006 ) = 2.33  radians . θ=cos −1 (−0.69006)=2.33 radians. Answer 2 The magnitudes of 𝐴 A and 𝐵 B: ∣ 𝐴 ∣ = 4.58 , ∣ 𝐵 ∣ = 3.16. ∣A∣=4.58,∣B∣=3.16. The angle: 𝜃 𝐴 𝐵 = cos ⁡ − 1 ( 𝐴 ⋅ 𝐵 ∣ 𝐴 ∣ ∣ 𝐵 ∣ ) θ AB ​ =cos −1 ( ∣A∣∣B∣ A⋅B ​ ) 𝜃 𝐴 𝐵 = cos ⁡ − 1 ( − 10 ( 4.58 ) ( 3.16 ) ) ≈ 13 4 ∘ . θ AB ​ =cos −1 ( (4.58)(3.16) −10 ​ )≈134 ∘ . Question 2 Find the maximum rate of change at the given point and the direction in which it occurs. 𝑓 ( 𝑥 , 𝑦 ) = 4 𝑦 𝑥 , ( 4 , 1 ) f(x,y)=4y x ​ ,(4,1). Answer The gradient vector gives the direction and magnitude of the maximum rate of change. Question 3 What is the position vector 𝑟 ( 𝑡 ) r(t) as a function of angle 𝜃 ( 𝑡 ) θ(t)? For uniform circular motion, find 𝜃 ( 𝑡 ) θ(t) at an arbitrary time 𝑡 t. Find 𝑟 r, the position vector, at time 𝑡 t. Determine an expression for the position vector of a particle that starts on the positive y-axis and moves with constant angular velocity 𝜔 ω. Answer Part A: 𝑟 ( 𝑡 ) = 𝑅 cos ⁡ ( 𝜃 ( 𝑡 ) ) 𝑖 + 𝑅 sin ⁡ ( 𝜃 ( 𝑡 ) ) 𝑗 . r(t)=Rcos(θ(t))i+Rsin(θ(t))j. Part B: 𝜃 ( 𝑡 ) = 𝜔 𝑡 . θ(t)=ωt. Part C: 𝑟 ( 𝑡 ) = 𝑅 cos ⁡ ( 𝜔 𝑡 ) 𝑖 + 𝑅 sin ⁡ ( 𝜔 𝑡 ) 𝑗 . r(t)=Rcos(ωt)i+Rsin(ωt)j. Part D: 𝑟 = 𝑅 𝑖 . r=Ri. Part E: The position vector for a particle starting on the y-axis: 𝑟 ( 𝑡 ) = − 𝑅 sin ⁡ ( 𝜔 𝑡 ) 𝑖 + 𝑅 cos ⁡ ( 𝜔 𝑡 ) 𝑗 . r(t)=−Rsin(ωt)i+Rcos(ωt)j. Question 4 Find the scalar and vector projections of 𝑏 b onto 𝑎 a. 𝑎 = ( 4 , 7 , − 4 ) , 𝑏 = ( 3 , − 1 , 1 ) a=(4,7,−4),b=(3,−1,1). Answer 1 Compute the dot product: 𝑎 ⋅ 𝑏 = ( 4 ) ( 3 ) + ( 7 ) ( − 1 ) + ( − 4 ) ( 1 ) = 12 − 7 − 4 = 1. a⋅b=(4)(3)+(7)(−1)+(−4)(1)=12−7−4=1. Find ∣ 𝑎 ∣ ∣a∣: ∣ 𝑎 ∣ = 4 2 + 7 2 + ( − 4 ) 2 = 16 + 49 + 16 = 81 = 9. ∣a∣= 4 2 +7 2 +(−4) 2 ​ = 16+49+16 ​ = 81 ​ =9. Scalar projection: comp 𝑎 𝑏 = 𝑎 ⋅ 𝑏 ∣ 𝑎 ∣ = 1 9 . comp a ​ b= ∣a∣ a⋅b ​ = 9 1 ​ . Vector projection: proj 𝑎 𝑏 = ( 𝑎 ⋅ 𝑏 ∣ 𝑎 ∣ 2 ) 𝑎 = ( 1 81 ) ( 4 , 7 , − 4 ) . proj a ​ b=( ∣a∣ 2 a⋅b ​ )a=( 81 1 ​ )(4,7,−4). proj 𝑎 𝑏 = ( 4 81 , 7 81 , − 4 81 ) . proj a ​ b=( 81 4 ​ , 81 7 ​ ,− 81 4 ​ ). Question 5 Find the vectors' orthogonality, parallelism, or lack thereof. 𝑢 = ( − 3 , 9 , 6 ) u=(−3,9,6), 𝑣 = ( 4 , − 12 , − 8 ) v=(4,−12,−8). Answer 1 Compute the dot product: 𝑢 ⋅ 𝑣 = ( − 3 ) ( 4 ) + ( 9 ) ( − 12 ) + ( 6 ) ( − 8 ) = − 12 − 108 − 48 = − 168. u⋅v=(−3)(4)+(9)(−12)+(6)(−8)=−12−108−48=−168. Compute the magnitudes: ∣ 𝑢 ∣ = ( − 3 ) 2 + 9 2 + 6 2 = 9 + 81 + 36 = 126 . ∣u∣= (−3) 2 +9 2 +6 2 ​ = 9+81+36 ​ = 126 ​ . ∣ 𝑣 ∣ = 4 2 + ( − 12 ) 2 + ( − 8 ) 2 = 16 + 144 + 64 = 224 . ∣v∣= 4 2 +(−12) 2 +(−8) 2 ​ = 16+144+64 ​ = 224 ​ . Check angle: cos ⁡ ( 𝜃 ) = 𝑢 ⋅ 𝑣 ∣ 𝑢 ∣ ∣ 𝑣 ∣ = − 168 126 224 . cos(θ)= ∣u∣∣v∣ u⋅v ​ = 126 ​ 224 ​ −168 ​ . Simplifying: cos ⁡ ( 𝜃 ) = − 1    ⟹    𝜃 = 𝜋 . cos(θ)=−1⟹θ=π. Conclusion: The vectors are parallel but point in opposite directions. Answer 2 The dot product confirms the vectors are not orthogonal ( 𝑢 ⋅ 𝑣 ≠ 0 u⋅v  =0). The magnitude comparison shows they are parallel. Answer 3 Direction ratios confirm proportionality: 𝑢 = ( − 3 , 9 , 6 ) and 𝑣 = ( 4 , − 12 , − 8 ) . u=(−3,9,6)andv=(4,−12,−8). Comparing direction ratios: − 3 4 = 9 − 12 = 6 − 8 = − 3 4 . 4 −3 ​ = −12 9 ​ = −8 6 ​ =− 4 3 ​ . The vectors are parallel. Answer 4 Steps confirm that the vectors are parallel because their direction ratios are proportional, and the dot product equals the product of their magnitudes.