Mathematical Analysis

      Section 12.1 Practice Problems • Problem 12.1.7: Describe and sketch the surface in ℝ3 represented by the equation 𝑥 + 𝑧 = 2. • Problem 12.1.11: Find the lengths of the given triangle 𝑃(3, −2, −3), 𝑄 7,0,1 , and 𝑅(1,2,1). Is it a right triangle or an isosceles triangle? • Problem 12.1.15: Find an equation of a sphere with center (−3,2,5) and radius 4. What is intersection of the sphere with the 𝑦𝑧-plane? Section 12.2 Practice Problems • Problem 12.2.20: For 𝑎 Ԧ = −3,4 and 𝑏 = 9, −1 , find 𝑎 Ԧ + 2𝑏, 𝑎 Ԧ , 𝑎 Ԧ − 𝑏 • Problem 12.2.24: Find a unit vector that has the same direction as −5,3, −1 Section 12.3 Practice Problems • Problem 12.3.6: Find the angle between the vectors 𝑎 Ԧ = 𝑖 Ԧ − 3𝑗 Ԧ and 𝑏 = −3𝑖 Ԧ + 4𝑗 Ԧ. • Problem 12.3.25 : Determine whether the triangle with the vertices 𝑃 1, −3, −2 ,𝑄(2,0, −4) and 𝑅(6, −2, −5) is a right-angled triangle. • Problem 12.3.27: Find a unit vector that is orthogonal to both 𝑖 Ԧ +𝑗 Ԧ and 𝑖 Ԧ + 𝑘. Section 12.4 Practice Problems • Exercise 12.4.11: By using the properties of cross products, determine the vector 𝑘 × (𝑖 − 2𝑗). • Exercise 12.4.19: Find two unit vectors orthogonal to both 3,2,1 and the vector −1,1,0 . • Exercise 12.4.27: Find the area of the parallelogram with the vertices 𝐴 −3,0 , 𝐵 −1,3 , 𝐶 5,2 , and 𝐷 3, −1 .