The distance between the point (5, 6, 4) and is
A) 4
B) 3
C) -4
D) -3

Question

The distance between the point (5, 6, 4) and  is
A) 4 
B) 3 
C) -4 
D) -3

Quizplus · Accepted Answer

Using the distance formula, we have: 
$\sqrt{(5-11e_{0})^{2}+(6-8e_{1})^{2}+(4-11e_{2})^{2}}$
From the answer choices, we can see that the distance is likely an integer value. Therefore, we can simplify the expression in the radical to look for perfect squares: 
$(5-11e_{0})^{2}+(6-8e_{1})^{2}+(4-11e_{2})^{2}$
$= 25 -110e_{0} +121e_{0}^{2}+36 -96e_{1} +64e_{1}^{2}+16 -88e_{2}+121e_{2}^{2}$
$= 169e_{0}^{2}+64e_{1}^{2}+121e_{2}^{2}-110e_{0}-96e_{1}-88e_{2}+81$
Simplifying modulo 5, we obtain:
$4e_{0}^{2}-e_{0}-e_{1}-3e_{2}+1\equiv 1\pmod{5}$
$4e_{0}^{2} - e_{0} \equiv e_{1} + 3e_{2} \pmod{5}$
We want the expression to be reduced into the form $a^{2}+b^{2}+c^{2},$ so we look for $e_{0}$ that makes the remaining two variables equal each other (or differ by a factor of 2). We find that substituting $e_{0}=1$ yields $3e_{2}\equiv e_{1}\equiv 3\pmod{5}.$ Substituting $e_{1}=3$ and $e_{2}=1$ into the original expression gives $||\overrightarrow{OP}||^{2}=9,$ so $||\overrightarrow{OP}||=\boxed{	extbf{(B)}\ 3}.$

The Distance Between the Point (5, 6, 4) and