# Thread: Parallelepiped volumes

1. ## Parallelepiped volumes

Hi there, pretty stuck on the following question from my exam revision:

Find the volume of the following parallelepiped, one of whose vertices is the origin, and with the edges linking the origin to the points (co-ordinates in Metres) (1) (1) (7)
(2) (0) (2)
(2) (3) (-1)

And also this, i think it is included:

Find the point of intersection of the following line and plane:
- The PLane passes through the points (3,3,4), (3,1,2) and (1,0,0)
- The line passes through the points (3,5,8) and (7,13,22)

THanks

2. Originally Posted by jameyt
Hi there, pretty stuck on the following question from my exam revision:

Find the volume of the following parallelepiped, one of whose vertices is the origin, and with the edges linking the origin to the points (co-ordinates in Metres) (1) (1) (7)
(2) (0) (2)
(2) (3) (-1)

And also this, i think it is included:

Find the point of intersection of the following line and plane:
- The PLane passes through the points (3,3,4), (3,1,2) and (1,0,0)
- The line passes through the points (3,5,8) and (7,13,22)

THanks
This will start you off:
Parallelepiped - Wikipedia, the free encyclopedia

3. Originally Posted by jameyt
Hi there, pretty stuck on the following question from my exam revision:

Find the volume of the following parallelepiped, one of whose vertices is the origin, and with the edges linking the origin to the points (co-ordinates in Metres) (1) (1) (7)
(2) (0) (2)
(2) (3) (-1)
The "triple" product of three vectors $(u \times v)\cdot w$ gives the volume of the parallelepiped having u, v, and w as edges. And that can be done most simply as a single determinant. If $u= $, $v= $, and $w= $ then $(u\times v)\cdot w= \left|\begin{array}{ccc}u_x & u_y & u_z \\ v_x & v_y & v_z \\ w_x & w_y & w_z\end{array}\right|$

And also this, i think it is included:

Find the point of intersection of the following line and plane:
- The PLane passes through the points (3,3,4), (3,1,2) and (1,0,0)
- The line passes through the points (3,5,8) and (7,13,22)

THanks
The vector from (3,3,4) to (3,1,2) is <0, -2, -2> and is a vector in the plane. The vector from (2,3,4) to (1, 0, 0) is <-1, -3, -4> and is another vector in the plane. Take the cross product of those two vectors to get a vector perpendicular to the plane. Can you write down the equation of a plane knowing one point and a perpendicular vector?

The vector from (3, 5, 8) to (7, 13, 22) is <-5, -8, -14>. Can you write parametric equations for a line given a direction vector and one point?

Put the parametric equations for x, y, and z into the equation for the plane and solve the equation for the parameter. The use that value to find the coordinates x, y, and z.