Answer:
The answer is
![\bold{ \left[\begin{array}{c}\frac{127}{27} \\ \ &\frac{65}{18} \\ \ &\frac{76}{27}\\ \ &\frac{97}{54}\\\ \end{array}\right]}](https://tex.z-dn.net/?f=%5Cbold%7B%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%5Cfrac%7B127%7D%7B27%7D%20%5C%5C%20%5C%20%26%5Cfrac%7B65%7D%7B18%7D%20%5C%5C%20%5C%20%26%5Cfrac%7B76%7D%7B27%7D%5C%5C%20%5C%20%26%5Cfrac%7B97%7D%7B54%7D%5C%5C%5C%20%5Cend%7Barray%7D%5Cright%5D%7D)
Step-by-step explanation:
Given value:
![v_1=\left[\begin{array}{c}5&4&3&2\\ \end{array}\right] \\](https://tex.z-dn.net/?f=v_1%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D5%264%263%262%5C%5C%20%5Cend%7Barray%7D%5Cright%5D%20%5C%5C)
![v_2=\left[\begin{array}{c}-4&5&-2&3\\ \end{array}\right] \\](https://tex.z-dn.net/?f=v_2%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D-4%265%26-2%263%5C%5C%20%5Cend%7Barray%7D%5Cright%5D%20%5C%5C)
The value of 
:
![= \left[\begin{array}{c}5&4&3&2\\ \end{array}\right] \left[\begin{array}{c}-4&5&-2&3\\ \end{array}\right]](https://tex.z-dn.net/?f=%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D5%264%263%262%5C%5C%20%5Cend%7Barray%7D%5Cright%5D%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D-4%265%26-2%263%5C%5C%20%5Cend%7Barray%7D%5Cright%5D)
The above value "
" are orthogonal vectors that is: 
from the above orthogonal basis of subspace w and the shortest distanc value of vector y:
![=\left[\begin{array}{c}3&9&-5&8\\\ \end{array}\right] \\\\](https://tex.z-dn.net/?f=%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D3%269%26-5%268%5C%5C%5C%20%5Cend%7Barray%7D%5Cright%5D%20%5C%5C%5C%5C)
The value of y on 
![= \frac{\left[\begin{array}{c}3&9&-5&8\\\ \end{array}\right] \cdot \left[\begin{array}{c}5&4&3&2\\ \end{array}\right] }{\left[\begin{array}{c}5&4&3&2\\ \end{array}\right] \cdot \left[\begin{array}{c}5&4&3&2\\ \end{array}\right]} \times \left[\begin{array}{c}5&4&3&2\\ \end{array}\right] +](https://tex.z-dn.net/?f=%3D%20%5Cfrac%7B%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D3%269%26-5%268%5C%5C%5C%20%5Cend%7Barray%7D%5Cright%5D%20%5Ccdot%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D5%264%263%262%5C%5C%20%5Cend%7Barray%7D%5Cright%5D%20%7D%7B%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D5%264%263%262%5C%5C%20%5Cend%7Barray%7D%5Cright%5D%20%5Ccdot%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D5%264%263%262%5C%5C%20%5Cend%7Barray%7D%5Cright%5D%7D%20%5Ctimes%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D5%264%263%262%5C%5C%20%5Cend%7Barray%7D%5Cright%5D%20%20%2B)
![\frac{\left[\begin{array}{c}3&9&5&-8\\\ \end{array}\right] \cdot \left[\begin{array}{c}-4&5&-2&3\\ \end{array}\right] }{\left[\begin{array}{c}-4&5&-2&3\\ \end{array}\right] \cdot \left[\begin{array}{c}-4&5&-2&3\\ \end{array}\right]} \times \left[\begin{array}{c}-4&5&-2&3\\ \end{array}\right]](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D3%269%265%26-8%5C%5C%5C%20%5Cend%7Barray%7D%5Cright%5D%20%5Ccdot%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D-4%265%26-2%263%5C%5C%20%5Cend%7Barray%7D%5Cright%5D%20%7D%7B%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D-4%265%26-2%263%5C%5C%20%5Cend%7Barray%7D%5Cright%5D%20%5Ccdot%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D-4%265%26-2%263%5C%5C%20%5Cend%7Barray%7D%5Cright%5D%7D%20%5Ctimes%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D-4%265%26-2%263%5C%5C%20%5Cend%7Barray%7D%5Cright%5D)
![= \frac{50}{54} \left[\begin{array}{c}5&4&3&2\\\ \end{array}\right] - \frac{1}{54} \left[\begin{array}{c}-4&5&-2&3\\\ \end{array}\right]\\\\](https://tex.z-dn.net/?f=%3D%20%5Cfrac%7B50%7D%7B54%7D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D5%264%263%262%5C%5C%5C%20%5Cend%7Barray%7D%5Cright%5D%20%20-%20%5Cfrac%7B1%7D%7B54%7D%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D-4%265%26-2%263%5C%5C%5C%20%5Cend%7Barray%7D%5Cright%5D%5C%5C%5C%5C)
![= \left[\begin{array}{c}\frac{127}{27} \\ \ &\frac{65}{18} \\ \ &\frac{76}{27}\\ \ &\frac{97}{54}\\\ \end{array}\right]](https://tex.z-dn.net/?f=%3D%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%5Cfrac%7B127%7D%7B27%7D%20%5C%5C%20%5C%20%26%5Cfrac%7B65%7D%7B18%7D%20%5C%5C%20%5C%20%26%5Cfrac%7B76%7D%7B27%7D%5C%5C%20%5C%20%26%5Cfrac%7B97%7D%7B54%7D%5C%5C%5C%20%5Cend%7Barray%7D%5Cright%5D)
The relationship is that 2x+3y=6 is 2* larger than 4x+6y=12
I'm not sure how to word the other two words but a prime number is a number that is only divisible by 1 and itself. Eg 13 is a prime as it is only divisible by 1 and itself.
