Mathematical Analysis

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

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