 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-2/11/a/ii/solution

BC

C A

and

A B

, respectively. The drawing places

E

F

and

G

on those edges and shows the arrows from the origin. The volume of their parallelepiped is the absolute scalar triple product:

e \times f = (\frac{1}{4}, \frac{1}{4}, - \frac{1}{4}), V = ∣ (e \times f) \cdot g ∣ = \frac{1}{4} .

Equivalently, this is the absolute determinant of the matrix whose columns are the three vectors. Since it is nonzero, the vectors are linearly independent. Three linearly independent vectors in the three-dimensional vector space

R^{3}

 New to topics? Read the docs here!