## Mathematical Analysis

Section 12.1 Practice Problems
β’ Problem 12.1.7: Describe and sketch the surface in β3
represented
by the equation π₯ + π§ = 2.
β’ Problem 12.1.11: Find the lengths of the given triangle π(3, β2, β3),
π 7,0,1 , and π(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 (β3,2,5)
and radius 4. What is intersection of the sphere with the π¦π§-plane?
Section 12.2 Practice Problems
β’ Problem 12.2.20: For π
Τ¦
= β3,4 and π = 9, β1 , find
π
Τ¦ + 2π, π
Τ¦
, π
Τ¦ β π
β’ Problem 12.2.24: Find a unit vector that has the same direction as
β5,3, β1
Section 12.3 Practice Problems
β’ Problem 12.3.6: Find the angle between the vectors π
Τ¦
= π
Τ¦ β 3π
Τ¦
and π = β3π
Τ¦ + 4π
Τ¦.
β’ Problem 12.3.25 : Determine whether the triangle with the vertices
π 1, β3, β2 ,π(2,0, β4) and π(6, β2, β5) is a right-angled triangle.
β’ Problem 12.3.27: Find a unit vector that is orthogonal to both π
Τ¦ +π
Τ¦
and π
Τ¦ + π.
Section 12.4 Practice Problems
β’ Exercise 12.4.11: By using the properties of cross products, determine
the vector π Γ (π β 2π).
β’ Exercise 12.4.19: Find two unit vectors orthogonal to both 3,2,1 and
the vector β1,1,0 .
β’ Exercise 12.4.27: Find the area of the parallelogram with the vertices
π΄ β3,0 , π΅ β1,3 , πΆ 5,2 , and π· 3, β1 .