Take the following prism:

[![question](https://i.stack.imgur.com/QQxuR.png)](https://i.stack.imgur.com/QQxuR.png)

To find its volume we first find the vector product of $\vec{OQ}$ and $\vec{OP}$:
$$\begin{bmatrix}i & j & k\\4 & -5 & 1\\3 & -3 & 1\end{bmatrix} = -2i -j + 3k$$

Then you find $\vec{QT}$:
$$\begin{bmatrix}4 \\-4 \\16 \end{bmatrix} - \begin{bmatrix}3 \\-3 \\1 \end{bmatrix} = \begin{bmatrix}1 \\-1 \\15 \end{bmatrix}$$

From there we perform a dot product operation:
$$\vec{QT}.\vec{OP}\*\vec{OQ} = 44$$

Then $44 / 2 = 22$ so the volume is $22 units^3$.

I follow this up until the dot product is used. My initial instinct when tackling this problem was to find the area of the triangle $OQP$ by performing $\frac{1}{2}|OP \* OQ|$, and then finding the length of the line $QT$ by calculating $\sqrt{(x\_1 - x\_2)^2 + (y\_1 - y\_2)^2 + (z\_1 - z\_2)^2}$, and then multiplying the area of the triangle $OQP$ by the length of line $QT$, but this gave me $28.2units^3$.

1) How/Why does the dot product find the volume of the cuboid (before you half it to get the prism volume)?

2) Why did my instinctual method not work?