Question

Find the volume of the parallelepiped determined by the vectors a, b, and c.

a = (1, 5, 2 ) b = (-1, 1, 4) c = (5, 1,2) in cubic units

Answer 1

To find the volume of the parallelepiped determined by vectors a, b, and c, you can use the triple scalar product. Calculate the cross product of b and c, then take the dot product of the cross product with a. Finally, divide the absolute value of the dot product by the magnitude of the cross product.

Step-by-step explanation:

To find the volume of the parallelepiped determined by the vectors a, b, and c, we can use the triple scalar product (also known as the scalar triple product). The triple scalar product is defined as the dot product of the cross product of two vectors with a third vector. In this case, we can calculate the volume as follows:

1. 
   
3. Calculate the cross product of vectors b and c: b x c = (-1, 1, 4) x (5, 1, 2) = (-6, -18, 6)
4. 
   
6. Take the dot product of the cross product with vector a: (-6, -18, 6) . (1, 5, 2) = -6 + (-90) + 12 = -84
7. 
   
9. The volume of the parallelepiped is given by the absolute value of the dot product divided by the magnitude of the cross product: |(-84)| / |(-6, -18, 6)| = 84 / sqrt(540) = 4 sqrt(15) cubic units