calc 3 homework

Homework #3 Due on: Tuesday May 5th (before noon). Part 1 – Vectors:

1. Determine whether the two vectors 𝑼 = 6𝒊 + 𝒋 + 3𝒌 , 𝑽 = 4𝒊 − 6𝒌

make an acute angle, obtuse angle, or are orthogonal. Explain your answer. 2. Let 𝑣 = 2𝒊 − 𝒋 + 3𝒌, 𝒃 = 𝒊 + 2𝒋 + 2𝒌 . First find the scalar projection of 𝒗 onto b (or, component of 𝒗 along b ) , then find the vector projection of 𝒗 onto b. 3. (a) Find the area of the parallelogram that has 𝒖 = 2𝒊 + 3𝒋 , 𝒗 = −𝒊 + 2𝒋 − 2𝒌 as its adjacent sides. (b) Find 𝒖. (𝒖 × 𝒗 ) (Explain your answer!) 4. (a) Study * Scalar Triple Product* on page 800 . Read carefully the explanations and example 5 . Then calculate 𝒖 . (𝒗 × 𝒘) where , 𝒖 = < 1 , −2 , 2 > , 𝒗 = < 0 , 3, 2 > , 𝒘 = < −4, 1 , −3 > . (b) Study *Theorem *11.4.6* on page 800. Part 2 – Lines and Planes 5. Find the parametric equations of the line that passes through the point (−2, 0 , 5) and is parallel to line given by 𝑥 = 𝑡, 𝑦 = −1 + 𝑡 , 𝑧 = 2. 6. Study problem 52 on page 812 , then do problem 54. 7.Find an equation of the plane that contains the line 𝑥 = 3𝑡 , 𝑦 = 1 + 𝑡 , 𝑧 = 2𝑡 and is parallel to the intersection of the planes 𝑦 + 𝑧 = −1 and 2𝑥 − 𝑦 + 𝑧 = 0. Part 3 – Quadratic Surfaces 8. Identify and sketch the given quadratic surface.

(a) 𝑧 = 𝑦2 − 𝑥2 (b) 9𝑥2 + 4𝑦2 − 36 𝑧2 = 0