Given \(\displaystyle -8i+2j-7k\), we need to map the \(\displaystyle i\), \(\displaystyle j\), and \(\displaystyle k\) coefficients back to their corresponding \(\displaystyle x\), \(\displaystyle y\), and \(\displaystyle z\)-coordinates.

Thus the vector form of \(\displaystyle -8i+2j-7k\) is \(\displaystyle \left \langle -8,2,-7\right \rangle\).

Given \(\displaystyle j+8k\), we need to map the \(\displaystyle i\), \(\displaystyle j\), and \(\displaystyle k\) coefficients back to their corresponding \(\displaystyle x\), \(\displaystyle y\), and \(\displaystyle z\)-coordinates.

Thus the vector form of \(\displaystyle j+8k\) is \(\displaystyle \left \langle 0,1,8\right \rangle\).

Given \(\displaystyle -i+3k\), we need to map the \(\displaystyle i\), \(\displaystyle j\), and \(\displaystyle k\) coefficients back to their corresponding \(\displaystyle x\), \(\displaystyle y\), and \(\displaystyle z\)-coordinates.

Thus the vector form of \(\displaystyle -i+3k\) is \(\displaystyle \left \langle -1,0,3\right \rangle\).

Given \(\displaystyle 7i-4j+k\), we need to map the \(\displaystyle i\), \(\displaystyle j\), and \(\displaystyle k\) coefficients back to their corresponding \(\displaystyle x\), \(\displaystyle y\), and \(\displaystyle z\)-coordinates.

Thus the vector form of \(\displaystyle 7i-4j+k\) is 

\(\displaystyle \left \langle 7,-4,1\right \rangle\).

Given \(\displaystyle 5i-2j+9k\), we need to map the \(\displaystyle i\), \(\displaystyle j\), and \(\displaystyle k\) coefficients back to their corresponding \(\displaystyle x\), \(\displaystyle y\), and \(\displaystyle z\)-coordinates.

Thus the vector form of \(\displaystyle 5i-2j+9k\) is 

\(\displaystyle \left \langle 5,-2,9\right \rangle\).

Given \(\displaystyle 2i-k\), we need to map the \(\displaystyle i\), \(\displaystyle j\), and \(\displaystyle k\) coefficients back to their corresponding \(\displaystyle x\), \(\displaystyle y\), and \(\displaystyle z\)-coordinates.

Thus the vector form of \(\displaystyle 2i-k\) is 

\(\displaystyle \left \langle 2,0,-1\right \rangle\).

The dot product of two vectors is the sum of the products of the vectors' corresponding elements. Given \(\displaystyle a=\left \langle 7,-7,0\right \rangle\) and \(\displaystyle b=\left \langle 2, -1,8\right \rangle\), then:

\(\displaystyle \left \langle 7,-7,0\right \rangle\times \left \langle 2, -1,8\right \rangle\)

\(\displaystyle =(7\times 2)+(-7\times -1)+(0\times 2)\)

\(\displaystyle =14+7+0\)

\(\displaystyle =21\) 

The dot product of two vectors is the sum of the products of the vectors' corresponding elements. Given \(\displaystyle a=\left \langle 4,1,-1\right \rangle\) and \(\displaystyle b=\left \langle 2, -1,8\right \rangle\), then:

\(\displaystyle \left \langle 4,1,-1\right \rangle\times \left \langle 2, -1,8\right \rangle\)

\(\displaystyle =(4\times 2)+(1\times -1)+(-1\times 8)\)

\(\displaystyle =8+(-1)+(-8)\)

\(\displaystyle =-1\) 

The dot product of two vectors is the sum of the products of the vectors' corresponding elements. Given \(\displaystyle a=\left \langle 3,0,-9\right \rangle\) and \(\displaystyle b=\left \langle -9, 2,-3\right \rangle\), then:

\(\displaystyle \left \langle 3,0,-9\right \rangle\times \left \langle -9, 2,-3\right \rangle\)

\(\displaystyle =(3\times -9)+(0\times 2)+(-9\times -3)\)

\(\displaystyle =(-27)+0+27\)

\(\displaystyle =0\) 

In order to derive the vector form, we must map the \(\displaystyle x\), \(\displaystyle y\), \(\displaystyle z\)-coordinates to their corresponding \(\displaystyle i\), \(\displaystyle j\), and \(\displaystyle k\) coefficients.

That is, given \(\displaystyle a=\left \langle a_{1},a_{2},a_{3}\right \rangle\), the vector form is \(\displaystyle a_{1}i+a_{2}j+a_{3}k\).

So for \(\displaystyle a=\left \langle 3,-2,7\right \rangle\), we can derive the vector form \(\displaystyle 3i-2j+7k\).