How do you find the surface area/volume of a rectangular box using vectors?
Considering vertexes at:
O: (0,0,0)
X: (9.5,0,0)
Y: (0,4.7,0)
Z: (0,0,3)
Do you need vectors? If you make a little sketch, you'd see that the rectangular box has dimensions of $\displaystyle 9.5 \times 4.7 \times 3$.
Volume is simply the product of all 3 dimensions. And the surface area is the sum of the area of each face of the box which you can figure out by simply looking at the dimensions of each face.
Well, I find this pretty trivial but ok ...
Let OX, OY, OZ be your vectors that define your paralleopiped (which happens to be a rectangular box). Thus, using the scalar triple product:
Volume = $\displaystyle \big| OX \cdot (OY \times OZ)\big| = \left| \text{det}\left[\begin{array}{ccc} 9.5 & 0 & 0 \\ 0 & 4.7 & 0 \\ 0 & 0 & 3 \end{array}\right] \right|$
which again is just multiplying the 3 dimensions.
For the surface area, you can use the fact that the area of a parallelogram (which happens to be a rectangle in this case) to be $\displaystyle \frac{1}{2} || \vec{u} \times \vec{v} ||$ and calculate the area of each face as I suggested.
Is it really that simple? I'm not sure why my teacher gave this for homework then...
Here's what it says:
INSTRUCTION: Measure the dimensions of a tissue box. Using one corner of the box as the origin, write the three vectors forming the adjacent edges. Use these vectors to find the surface area and the volume of the box. Vector calculations are to be done on a piece of paper, NOT the tissue box. Credit will be awarded only if the calculations showing all set up and work are accompanied by a full box of tissues.