## Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

$\sqrt {38}-\sqrt {17}$
Let us consider two vectors $v=pi+qj+rz$ and $w=xi+yj+zk$ The addition of the above vectors can be found by adding the individual components of the vector: $v+w=(p+x)i+(q+y)j+(r+z)k$ We also know that the magnitude of any vector (let us say $v$) can be determined using the formula $||v||=\sqrt{p^2+q^2+r^2}$ We apply similar reasoning to simplify the given expression: Therefore, $||v || -||w||=|| (3i-5j+2k) || -|| (-2i+3j-2k)|| \\ = \sqrt {(3)^2+(-5)^2+(2)^2} -\sqrt {(-2)^2+(3)^2+(-2)^2} \\=\sqrt {9+25+4}-\sqrt {4+9+4} \\ = \sqrt {38}-\sqrt {17}$