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 .