Area and volume with the cross product
The length of the cross product of two vectors is the area of their parallelogram. With three vectors, a determinant gives the volume of the parallelepiped.
📖 Start here: read the steps of the “Lesson” in order (1, 2, 3 …) and answer each question. Then continue with “Practice”.
- Iran: grade 12
- Russia: grade 11
Guide
This studio continues the dot and cross product section of Iranian Year 12 geometry 3, chapter 3. If a and b are two sides of a parallelogram, its area is the length of their vector product (cross product), |a×b|. A triangle on the same two vectors has half that area.
Three vectors that do not lie in one plane make a parallelepiped. Its volume is |a·(b×c)|. This is the absolute value of the 3 × 3 determinant whose rows are the components of a, b and c. If the determinant is zero the volume is zero: the three vectors lie in one plane.
Common mistakes: forgetting to halve for a triangle, forgetting the minus sign of the middle term when expanding the determinant, and adding the components instead of finding the vector's length.