This small group activity is designed to help students visualize the cross product. Students work in small groups to determine the area of a triangle in space. The whole class wrap-up discussion emphasizes the geometric interpretation of the cross product.
Find the area of the part of the plane \(x+y+1\) in the first octant, as shown in the figure beolw.
- Students may have difficulty representing the sides of the triangle as vectors. Encouraging them to represent the corners as vectors may help.
- Students may have difficulty visualizing the triangle as half of a parallelogram -- and may forget the factor of \(\frac12\) even once they recognize the need for the cross product.
- Some students will solve this problem geometrically, without vectors.