# If A= <7 ,-5 ,1 > and B= <28 ,6 ,7 >, what is A*B -||A|| ||B||?

$173 - 5 \sqrt{2607}$
For a vector $\vec{X} = \left(\begin{matrix}{x}_{1} \\ {x}_{2} \\ {x}_{3}\end{matrix}\right)$, its magnitude is $| | \vec{X} | | = \sqrt{{x}_{1}^{2} + {x}_{2}^{2} + {x}_{3}^{2}}$. For two vectors $\vec{X} = \left(\begin{matrix}{x}_{1} \\ {x}_{2} \\ {x}_{3}\end{matrix}\right)$ and $\vec{Y} = \left(\begin{matrix}{y}_{1} \\ {y}_{2} \\ {y}_{3}\end{matrix}\right)$, their dot product is $\vec{X} \cdot \vec{Y} = {x}_{1} {y}_{1} + {x}_{2} {y}_{2} + {x}_{3} {y}_{3}$.
From the above, $\vec{A} \cdot \vec{B} = 7 \cdot 28 + - 5 \cdot 6 + 1 \cdot 7 = 173$. The magnitude of $\vec{A} = | | \vec{A} | | = \sqrt{{7}^{2} + {\left(- 5\right)}^{2} + {1}^{2}} = \sqrt{75} = 5 \sqrt{3}$. The magnitude of $\vec{B} = | | \vec{B} | | = \sqrt{{28}^{2} + {6}^{2} + {7}^{2}} = \sqrt{869}$.
Therefore, $\vec{A} \cdot \vec{B} - | | \vec{A} | | | | \vec{B} | | = 173 - 5 \sqrt{3} \sqrt{869} = 173 - 5 \sqrt{2607}$.