Question

Find the length of the parallelogram determined by \(u=<2,4,-1>\) and \(v=<0,-3,2>\).

Short answer

The lengths of the diagonals are \(\sqrt{6}\) and \(\sqrt{62}\).

Step-by-step solution

Given Information

The given vectors are \(u=<2,4,-1>\) and \(v=<0,-3,2>\).

The lengths of the diagonals are:

\(d_1=|u+v|\) and \(d_2=|u-v|\).

Find the lengths of the diagonals

The length of the 1st diagonal is,

\(\begin{align}d_1&=|(2i+4j-k)+(0i-3j+2k)|\\&=|2i+j+k|\\&=\sqrt{2^2+1^2+1^2}\\&=\sqrt{6}\end{align}\).

The length of the 2nd diagonal is,

\(\begin{align}d_2&=|(2i+4j-k)-(0i-3j+2k)|\\&=|2i+7j-3k|\\&=\sqrt{2^2+7^2+(-3)^2}\\&=\sqrt{4+49+9}\\&=\sqrt{62}\end{align}\).