# How do you find the volume of the parallelepiped with adjacent edges pq, pr, and ps where p(3,0,1), q(-1,2,5), r(5,1,-1) and s(0,4,2)?

Given three vectors, there is a product, called scalar triple product, that gives (the absolute value of it), the volume of the parallelepiped that has the three vectors as dimensions.

So:

and the derminant is given for example with the Laplace rule (choosing the first row):

By signing up, you agree to our Terms of Service and Privacy Policy

To find the volume of the parallelepiped with adjacent edges ( \overrightarrow{pq} ), ( \overrightarrow{pr} ), and ( \overrightarrow{ps} ), where ( p(3,0,1) ), ( q(-1,2,5) ), ( r(5,1,-1) ), and ( s(0,4,2) ), you can use the scalar triple product formula:

[ V = | \overrightarrow{pq} \cdot (\overrightarrow{pr} \times \overrightarrow{ps}) | ]

Where ( \overrightarrow{pq} ) is the vector from ( p ) to ( q ), ( \overrightarrow{pr} ) is the vector from ( p ) to ( r ), and ( \overrightarrow{ps} ) is the vector from ( p ) to ( s ).

The cross product ( \overrightarrow{pr} \times \overrightarrow{ps} ) can be calculated using the determinant of a matrix:

[ \overrightarrow{pr} \times \overrightarrow{ps} = \begin{pmatrix} i & j & k \ x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \ x_3 - x_1 & y_3 - y_1 & z_3 - z_1 \end{pmatrix} ]

Where ( (x_1, y_1, z_1) ), ( (x_2, y_2, z_2) ), and ( (x_3, y_3, z_3) ) are the coordinates of points ( p ), ( r ), and ( s ) respectively.

Once you have the cross product, you can then calculate the dot product with ( \overrightarrow{pq} ), take the absolute value, and find the volume of the parallelepiped.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find the volume of the solid generated when the regions bounded by the graphs of the given equations #y = 2/sqrtx, x=1, x=5# and the #x#-axis are rotated about the #x#-axis?
- How do you find the volume of the wedge-shaped region on the figure contained in the cylinder #x^2 + y^2 = 16# and bounded above by the plane #z = x# and below by the xy-plane?
- How do you find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane #x + 6y + 10z = 60#?
- How do you Find the volume of the solid that lies in the first octant and is bounded by the three coordinate planes and another plane passing through (3,0,0), (0,4,0), and (0,0,5)?
- A solid has a circular base of radius 1. It has parallel cross-sections perpendicular to the base which are equilateral triangles. How do you find the volume of the solid?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7