MediumMathematicsHigh School (11–12)Vectors and Three-Dimensional GeometryNumerical

A concrete block leans to one side. Its three edges from one corner are a = 3 i + j - k, b = i + 2…

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

A concrete block leans to one side. Its three edges from one corner are $\vec a = 3\hat i + \hat j - \hat k$ , $\vec b = \hat i + 2\hat j + \hat k$ and $\vec c = 2\hat i - \hat j + 4\hat k$ , in decimetres. Find the volume of the block. The same three edges also describe a tetrahedral wedge, which is cut from the block with its corner at the common point. Find the volume of that wedge.

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.

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