Question

Find the volume of the parallelepiped determined by the vectors a, b, and c. a = 4, 3, −1 , b = 0, 1, 2 , c = 5, −3, 6

Answer 1

A parallelepiped is a three-dimensional object, with six equal faces of pairs, which is formed by intercepting and prolonging the three base vectors that make it up, with which each face or parallelogram is formed.

To calculate the volume of the parallelepiped, use is made of the triple product, | a. (Bxc) |, which combines the scalar and vector product, in addition it is equivalent to the determinate of the 3x3 matrix formed by the 3 vectors, so in this case we have:

`det\left[\begin{array}{ccc}4&3&-1\\0&1&2\\5&-3&6\end{array}\right] =(4)det\left[\begin{array}{cc}1&2\\-3&6\end{array}\right] -(3)det\left[\begin{array}{cc}0&2\\5&6\end{array}\right]+(-1)det\left[\begin{array}{cc}0&1\\5&-3\end{array}\right]\\  =4[6-(-6)]-3(0-10)-(0-5)=48+30+5=83`

Answer

The volume of the described parallelepiped is 83 cubic units