Four joints of a steel frame have the coordinates A(1, 2, 3), B(2, 4, 5), C(0, 1, 1) and D(2, 2…

Contributed by StemPy Lab

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

Four joints of a steel frame have the coordinates $A(1, 2, 3)$ , $B(2, 4, 5)$ , $C(0, 1, 1)$ and $D(2, 2, \lambda)$ , in metres. The designer wants the four joints in one flat plane, so that a single panel can cover them. Find the value of $\lambda$ for which this holds.

Need a reminder? The key ideaShow

$[\vec a\ \vec b\ \vec c] = \vec a\cdot(\vec b\times\vec c)$ is the determinant of the three vectors. The parallelepiped has volume $\lvert[\vec a\ \vec b\ \vec c]\rvert$ and the tetrahedron has $\dfrac{1}{6}$ of that. The product is zero when the vectors are coplanar.

The worked solution

Have you tried it?

Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

← All Problems

← A concrete block leans to one side. Its three edges from one corner are a = 3 i + j - k, b = i + 2…A tent is a tetrahedron with the corners P(1, 0, 0), Q(0, 2, 0), R(0, 0, 3) and S(1, 1, 1), with… →

More onVectors and Three-Dimensional Geometry › Vector Algebra

Discussion (0)

Loading…