University:
Johns Hopkins University
Course:
AS.110.601 | Algebra I
Academic year:

2024
Views:

294

Pages:

8
Author:

Cristian Santos

Question 1 Find the projection of u onto v and v onto u. Given: 𝑢 = ⟨ − 2 , 6 , 4 ⟩ u=⟨−2,6,4⟩, 𝑣 = 𝜆 ⟨ 2 , 4 , 4 ⟩ v=λ⟨2,4,4⟩, Projection of 𝑢 u onto 𝑣 v is ⟨ 2 , 4 , 4 ⟩ ⟨2,4,4⟩, Vector projection of 𝑣 v onto 𝑢 u is ⟨ − 8 , 24 , 16 ⟩ ⟨−8,24,16⟩. Steps: Projection formula for vector u onto vector v: proj 𝑣 𝑢 = 𝑢 ⋅ 𝑣 ∣ 𝑣 ∣ 2 𝑣 . proj v u= ∣v∣ 2 u⋅v v. Solving for 𝜆 λ: 𝑢 ⋅ 𝑣 = − 2 ( 2 ) + 6 ( 4 ) + 4 ( 4 ) = − 4 + 24 + 16 = 36 u⋅v=−2(2)+6(4)+4(4)=−4+24+16=36, Magnitude of 𝑣 v: ∣ 𝑣 ∣ 2 = ( 2 ) 2 + ( 4 ) 2 + ( 4 ) 2 = 4 + 16 + 16 = 36. ∣v∣ 2 =(2) 2 +(4) 2 +(4) 2 =4+16+16=36. Substituting: proj 𝑣 𝑢 = 36 36 𝑣 = 𝑣 . proj v u= 36 36 v=v. Since the projection matches, we confirm: 𝑣 = 𝜆 ⟨ 2 , 4 , 4 ⟩ = 56 9 ⟨ 2 , 4 , 4 ⟩ . v=λ⟨2,4,4⟩= 9 56 ⟨2,4,4⟩. Question 2 Which expressions are meaningful or meaningless? Answers: (a) Meaningless: Dot product ( 𝑎 ⋅ 𝑏 ) (a⋅b) is scalar, and scalar cannot dot with vector c. (b) Meaningful: ( 𝑎 ⋅ 𝑏 ) 𝑐 (a⋅b)c is scalar multiplication of c. (c) Meaningful: ∣ 𝑎 ∣ ( 𝑏 ⋅ 𝑐 ) ∣a∣(b⋅c) is multiplication of scalars. (d) Meaningful: ( 𝑎 ⋅ ( 𝑏 + 𝑐 ) ) (a⋅(b+c)) is a dot product of two vectors. (e) Meaningless: 𝑎 ⋅ 𝑏 + 𝑐 a⋅b+c adds scalar to vector. (f) Meaningless: ∣ 𝑎 ∣ ( 𝑏 ⋅ 𝑐 ) ∣a∣(b⋅c) has scalar and vector dot product. Question 3 Find angles of the triangle with vertices A(1,0,-1), B(3,-2,0), C(1,3,3): Find vectors: 𝐴 𝐵 = ⟨ 2 , − 2 , 1 ⟩ , 𝐴 𝐶 = ⟨ 0 , 3 , 4 ⟩ , 𝐵 𝐶 = ⟨ − 2 , 5 , 3 ⟩ . AB=⟨2,−2,1⟩,AC=⟨0,3,4⟩,BC=⟨−2,5,3⟩. Magnitudes: ∣ 𝐴 𝐵 ∣ = 2 2 + ( − 2 ) 2 + 1 2 = 9 = 3 , ∣ 𝐴 𝐶 ∣ = 3 2 + 4 2 = 5. ∣AB∣= 2 2 +(−2) 2 +1 2 = 9 =3,∣AC∣= 3 2 +4 2 =5. Angles using dot product: 𝐴 𝐵 ⋅ 𝐴 𝐶 = ( 2 ) ( 0 ) + ( − 2 ) ( 3 ) + ( 1 ) ( 4 ) = − 6 + 4 = − 2 AB⋅AC=(2)(0)+(−2)(3)+(1)(4)=−6+4=−2. cos ⁡ 𝜃 = 𝐴 𝐵 ⋅ 𝐴 𝐶 ∣ 𝐴 𝐵 ∣ ∣ 𝐴 𝐶 ∣ = − 2 3 ⋅ 5 = − 2 15 . cosθ= ∣AB∣∣AC∣ AB⋅AC = 3⋅5 −2 =− 15 2 . Solve for angles: 𝜃 = arccos ⁡ ( − 2 15 ) . θ=arccos(− 15 2 ). Repeat similar steps for other vectors. Final approximate angles: 𝛼 ≈ 46.5 7 ∘ α≈46.57 ∘ , 𝛽 ≈ 58.9 7 ∘ β≈58.97 ∘ , 𝛾 ≈ 73.9 5 ∘ γ≈73.95 ∘ . Question 4 Vectors 𝑉 1 V 1 and 𝑉 2 V 2 : 𝑉 1 ⋅ 𝑉 1 V 1 ⋅V 1 : 𝑉 1 ⋅ 𝑉 1 = ∣ 𝑉 1 ∣ 2 . V 1 ⋅V 1 =∣V 1 ∣ 2 . 𝑉 1 ⋅ 𝑉 2 V 1 ⋅V 2 when perpendicular: If perpendicular, 𝜃 = 9 0 ∘ θ=90 ∘ : 𝑉 1 ⋅ 𝑉 2 = ∣ 𝑉 1 ∣ ∣ 𝑉 2 ∣ cos ⁡ 9 0 ∘ = 0. V 1 ⋅V 2 =∣V 1 ∣∣V 2 ∣cos90 ∘ =0. 𝑉 1 ⋅ 𝑉 2 V 1 ⋅V 2 when parallel: If parallel, 𝜃 = 0 ∘ θ=0 ∘ : 𝑉 1 ⋅ 𝑉 2 = ∣ 𝑉 1 ∣ ∣ 𝑉 2 ∣ . V 1 ⋅V 2 =∣V 1 ∣∣V 2 ∣. Question 5 Find the curve of intersection of 𝑥 2 + 𝑦 2 = 4 x 2 +y 2 =4 and 𝑧 = 𝑥 𝑦 z=xy: Parametrize 𝑥 x and 𝑦 y: 𝑥 = 2 cos ⁡ 𝑡 , 𝑦 = 2 sin ⁡ 𝑡 . x=2cost,y=2sint. Substitute into 𝑧 = 𝑥 𝑦 z=xy: 𝑧 = ( 2 cos ⁡ 𝑡 ) ( 2 sin ⁡ 𝑡 ) = 4 sin ⁡ 𝑡 cos ⁡ 𝑡 . z=(2cost)(2sint)=4sintcost. Final vector function: 𝑟 ( 𝑡 ) = ⟨ 2 cos ⁡ 𝑡 , 2 sin ⁡ 𝑡 , 4 sin ⁡ 𝑡 cos ⁡ 𝑡 ⟩ . r(t)=⟨2cost,2sint,4sintcost⟩.

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