Question

Compute the lengths of the four diagonals of the parallelepiped

determined by \(u=i\), \(v=2j\), and \(w=2k\).

Short answer

The lengths of the four diagonals are \(5,5,2\sqrt2,\text{ and }3\).

Step-by-step solution

Given Information

A parallelepiped formed by \(u=i\), \(v=2j\), and \(w=2k\).

There are four diagonals three faces and one body diagonal.

Three faces diagonals are:

\(d_1=u+v\)

\(d_2=u+w\)

\(d_3=v+w\)

One body diagonal is:

\(d_4=u+v+w\)

Calculate the lengths of the diagonals

The length of 1st face diagonal:

\(d_1=|i+2j|\)

\(d_1=\sqrt{1+4}\)

\(d_1=\sqrt5\)

The length of 2nd face diagonal:

\(d_2=|i+2k|\)

\(d_2=\sqrt{1+4}\)

\(d_2=\sqrt5\)

The length of 3rd face diagonal:

\(d_3=|2j+2k|\)

\(d_3=\sqrt{4+4}\)

\(d_3=2\sqrt2\)

The length of the body diagonal:

\(d_4=|i+2j+2k|\)

\(d_4=\sqrt{1+4+4}\)

\(d_4=3\)