Home»Math Guides»Finding Volume of Parallelpiped using Triple Product
Volume of a Parallelpiped 3D Shape – Using the Triple Product
In linear algebra courses you may be asked to find the volume of a parallelpiped, and you’ll be given either points of vertices or vectors that make up the parallelpiped.
Here’s an example of finding the volume of a parallelpiped, which can be done either by the triple product or a 3×3 determinant.
Find the volume of the parallelpiped formed from the points P(3,0,1), Q(-1,2,5), R(5,1,-1) and S(0,4,2).
The first step is to turn the points into vectors.
It’s best to make all the vectors from the same point when finding the volume.
For example, we can find the vectors PQ, PR and PS and then use the triple product to find the volume.
You can also write the vectors with respect to another point as well.
Remember that the vector is made by subtracting the second letter point from the first letter point.
To find PQ for example, subtract point P from point Q..
\[\mathop {PQ}\limits^ \to = \left\langle { – 1 – 3,2 – 0,5 – 1} \right\rangle \]
\[\mathop {PQ}\limits^ \to = \left\langle { – 4,2,4} \right\rangle \]
And repeat for the other vectors and points.
\[\mathop {PR}\limits^ \to = \left\langle {5 – 3,1 – 0, – 1 – 1} \right\rangle \]
\[\mathop {PR}\limits^ \to = \left\langle {2,1, – 2} \right\rangle \]
\[\mathop {PS}\limits^ \to = \left\langle {0 – 3,4 – 0,2 – 1} \right\rangle \]
\[\mathop {PS}\limits^ \to = \left\langle { – 3,4,1} \right\rangle \]
Let a, b and c be the vectors PQ, PR and PS, respectively.
Finding the volume can be done from either the triple product \(\left| {a \cdot (b \times c)} \right|\) or by just putting the three vectors into a 3×3 matrix and finding the determinant.
Let’s use the triple product method first.
Note that those are absolute signs around the triple product, it’s NOT the norm.
This is because the way you choose the a b and c vectors may cause you to get a negative of the answer, just take the positive of it!
Let b and c be \(\left\langle {2,1,- 2} \right\rangle \) and \(\left\langle { – 3,4,1} \right\rangle \), respectively.
To find the cross product of b and c, write them into a matrix, and find each component by covering up the column you want to find with you hand and finding the determinant of the remaining 2×2 matrix, and make sure the middle term is multiplied by negative 1.
\[\begin{array}{*{20}{c}}i&j&k\\2&1&{ – 2}\\{ – 3}&4&1\end{array}\]
- For the first component for example, you put your hand over the first column, and find the determinant of what remains.
- Same with the second component, but after finding the determinant multiply it by negative one because of the cofactor matrix
- Repeat for the third component but don’t multiply that by a negative. See our worked through calculations:
\[\left( {b \times c} \right) = \left\langle {(1)(1) – ( – 2)(4), – [(2)(1) – ( – 2)( – 3)],(2)(4) – (1)( – 3)} \right\rangle \]
\[\left( {b \times c} \right) = \left\langle {9,4,11} \right\rangle \]
And now just find the dot product part of the triple product, which is just multiplying both components by each other and summing it up.
\[\left| {a \cdot \left( {b \times c} \right)} \right| = \left\langle { – 4,2,4} \right\rangle \cdot \left\langle {9,4,11} \right\rangle \]
\[\left| {a \cdot \left( {b \times c} \right)} \right| = ( – 4)(9) + (2)(4) + (4)(11)\]
\[\left| {a \cdot \left( {b \times c} \right)} \right| = 16\]
And we found the volume, 16 units cubed!
What about the other way, by putting all the vectors into a matrix and finding the determinant? Let’s do that:
\[\left| {\begin{array}{*{20}{c}}{ – 4}&2&4\\2&1&{ – 2}\\{ – 3}&4&1\end{array}} \right|\]
And just find the determinant of the 3×3 matrix, which just involves making multiple 2×2 determinants by covering the row and column of the piece you are working with.
Note that the middle part term is negative.
\[V = – 4[(1)(1) – ( – 2)(4)] – 2[(2)(1) – ( – 2)( – 3)] + 4[(2)(4) – (1)( – 3)]\]
\[V = 16\]
It will be in units cubed, so if you were given units in the question statement, put those units cubed.
Double checking the triple product (cross product and dot product) using MATLAB
You can double check your results by using a program called MATLAB, which specializes in using arrays (vectors and matrices), use the following code script m-file and feel free to replace the numbers with the numbers for your vectors to find your triple product:
%bai-gaming.com/math-guides
PQ = [-4,2,4]
PR = [2,1,-2]
PS = [-3,4,1]
dot(PQ,(cross(PR,PS)))And you will acquire a result of 16, great!
Click here to see another topic, how to find partial derivative for multivariable functions!