Question 1 Black vultures glide along a path tipped 3.5° below the horizontal. If the horizontal distance is 100 m, how much height does it lose? Answer: The vertical distance ( 𝑋 𝑣 X v ) can be found using the equation: 𝑋 𝑣 = 𝑑 ℎ ⋅ tan ⁡ ( 𝜃 ) , X v =d h ⋅tan(θ), where 𝑑 ℎ = 100 d h =100 m and 𝜃 = 3. 5 ∘ θ=3.5 ∘ . Substituting: 𝑋 𝑣 = 100 ⋅ tan ⁡ ( 3. 5 ∘ ) ≈ 100 ⋅ 0.061 = 6.11 m . X v =100⋅tan(3.5 ∘ )≈100⋅0.061=6.11m. Question 2 Find one eigenvalue and two linearly independent eigenvectors of: 𝐴 = [ 5 5 5 5 5 5 5 5 5 ] . A= 5 5 5 5 5 5 5 5 5 . Answer: The matrix 𝐴 A is not invertible because its columns are linearly dependent. Hence, 0 0 is an eigenvalue. Two linearly independent eigenvectors correspond to any two linearly independent solutions to 𝐴 𝑣 = 0 Av=0. Question 3 Find the angle between the given vector 𝑣 = 𝑖 + 3 𝑗 v=i+ 3 j and the positive 𝑥 x-axis. Answer: The dot product of 𝑣 v and 𝑖 i: 𝑣 ⋅ 𝑖 = 1 ⋅ 1 + 3 ⋅ 0 = 1. v⋅i=1⋅1+ 3 ⋅0=1. Magnitude of 𝑣 v: ∣ 𝑣 ∣ = 1 2 + ( 3 ) 2 = 4 = 2. ∣v∣= 1 2 +( 3 ) 2 = 4 =2. Cosine of the angle: cos ⁡ ( 𝜃 ) = 𝑣 ⋅ 𝑖 ∣ 𝑣 ∣ ⋅ ∣ 𝑖 ∣ = 1 2 . cos(θ)= ∣v∣⋅∣i∣ v⋅i = 2 1 . 𝜃 = cos ⁡ − 1 ( 0.5 ) = 6 0 ∘ θ=cos −1 (0.5)=60 ∘ . Question 4 Find a unit vector in the same direction as 𝑣 = 8 𝑖 − 𝑗 + 4 𝑘 v=8i−j+4k. Answer: Magnitude of 𝑣 v: ∣ 𝑣 ∣ = 8 2 + ( − 1 ) 2 + 4 2 = 64 + 1 + 16 = 81 = 9. ∣v∣= 8 2 +(−1) 2 +4 2 = 64+1+16 = 81 =9. Unit vector: 𝑢 = 𝑣 ∣ 𝑣 ∣ = 1 9 ( 8 𝑖 − 𝑗 + 4 𝑘 ) = 8 9 𝑖 − 1 9 𝑗 + 4 9 𝑘 . u= ∣v∣ v = 9 1 (8i−j+4k)= 9 8 i− 9 1 j+ 9 4 k. Question 5 Identify the surface for 𝑟 ( 𝑠 , 𝑡 ) = ⟨ 𝑠 , 𝑡 , 𝑡 2 − 𝑠 2 ⟩ r(s,t)=⟨s,t,t 2 −s 2 ⟩. Answer: The parametric equations: 𝑥 = 𝑠 , 𝑦 = 𝑡 , 𝑧 = 𝑡 2 − 𝑠 2 ⟹ 𝑧 = 𝑦 2 − 𝑥 2 . x=s,y=t,z=t 2 −s 2 ⟹z=y 2 −x 2 . This is the equation of a hyperbolic paraboloid.