Vector problem on finding point on the line

[jaychay]

Find point d on the line of r(t)=(0,0,0)+(−1,1,1)t which make the triangular pyramid abcd has the volume of 4 unit cube when a(0,0,0),b(1,0,1),c(0,1,0) are the points on the plane of −x+z=0.

[shyamjayakannan]

volume = [tex]\frac{\left|\left(\vec{ab}\times\vec{ac}\right)\cdot\vec{ad}\right|}{6}[/tex]. Now, [tex]\vec{ab}=\langle1,0,1\rangle[/tex], [tex]\vec{ac}=\langle0,1,0\rangle[/tex] and [tex]\vec{ad}=\langle-t,t,t\rangle[/tex].

So, [tex]\frac{\left|\left(\vec{ab}\times\vec{ac}\right)\cdot\vec{ad}\right|}{6}=4\Rightarrow\frac{1}{6}\left|\left|\begin{matrix}1&0&1\\0&1&0\\-t&t&t\end{matrix}\right|\right|=4\Rightarrow\frac{|2t|}{6}=4\Rightarrow|t|=12\Rightarrow t= \pm 12[/tex]

There are two points, one above and one below the plane, which will give the same volume. From the diagram, it looks like the point they want is for [tex]\boxed{t=12}[/tex].