Test on Scalar triple product and the volume of a parallelepiped

Question 1: The volume of a parallelepiped formed by three non-coplanar vectors a, b, and c is directly represented by the scalar triple product (a × b)·c.

A. Yes

B. No

Explanation: The study materials state that the absolute value of the scalar triple product, |(a × b)·c|, gives the volume of the parallelepiped. The scalar triple product (a × b)·c can be negative depending on the orientation of the vectors, while volume is always a non-negative quantity, hence the need for the absolute value.

Question 2: Given three vectors a = (a₁, a₂, a₃), b = (b₁, b₂, b₃), and c = (c₁, c₂, c₃), which of the following expressions correctly represents the scalar triple product (a × b)·c using a determinant?

A. | a₁ a₂ a₃ |\n| b₁ b₂ b₃ |\n| c₁ c₂ c₃ |

B. | a₁ b₁ c₁ |\n| a₂ b₂ c₂ |\n| a₃ b₃ c₃ |

C. | a₁ a₂ a₃ |\n| c₁ c₂ c₃ |\n| b₁ b₂ b₃ |

D. | b₁ b₂ b₃ |\n| a₁ a₂ a₃ |\n| c₁ c₂ c₃ |

Explanation: The study materials state that the scalar triple product (a × b)·c is equal to the determinant of the matrix formed by these vectors as rows or columns. This means the vectors can be arranged as rows or as columns. Therefore, representing the vectors a, b, and c as rows, or as columns, both correctly represent the scalar triple product.

Question 3: The volume of a parallelepiped formed by three vectors is calculated by taking the absolute value of their scalar triple product.

A. Yes

B. No

Explanation: The study materials state: 'The volume V of the parallelepiped is the absolute value of the scalar triple product. V = |(a × b)·c|'.

Question 4: To avoid sign errors when calculating a 3x3 determinant for the scalar triple product, is it advisable to double-check each term?

A. Yes

B. No

Explanation: The study materials state that 'Careless calculation of the 3x3 determinant, especially with negative signs, can lead to incorrect results. Double-checking each term is advisable.' This directly supports that double-checking each term is advisable to avoid sign errors.

Question 5: Which of the following expressions correctly represents the volume of a tetrahedron (V₄) formed by three vectors a, b, and c originating from a common vertex?

A. V₄ = (a × b)·c / 6

B. V₄ = |(a × b)·c| / 6

C. V₄ = |(a × b)·c|

D. V₄ = (a · b) × c / 6

Explanation: The study materials state that if three vectors a, b, and c originate from a common vertex of a tetrahedron, its volume (V₄) is given by the formula V₄ = |(a × b)·c| / 6. The absolute value is crucial because volume must always be non-negative. (a × b)·c represents the scalar triple product, and its absolute value is the volume of the parallelepiped formed by the vectors; the tetrahedron's volume is one-sixth of that.