Given the vectors (\vec{OA} = \begin{pmatrix} -6 \ 8 \end{pmatrix}) and (\vec{OB} = \begin{pmatrix} -14 \ -16 \end{pmatrix}), with (AB : AN = 4:1):Find the coordinates of point (N).?

To find the coordinates of point N, we first need to find the vector \vec{AB} by subtracting the coordinates of point A from the coordinates of point B:

\vec{AB} = \begin{pmatrix} -14 \\ -16 \end{pmatrix} - \begin{pmatrix} -6 \\ 8 \end{pmatrix} = \begin{pmatrix} -8 \\ -24 \end{pmatrix}

Since the ratio of AB to AN is 4:1, we can find the coordinates of point N by dividing the coordinates of point B by 4 and adding it to the coordinates of point A:

\vec{AN} = \frac{1}{4} \vec{AB} = \frac{1}{4} \begin{pmatrix} -8 \\ -24 \end{pmatrix} = \begin{pmatrix} -2 \\ -6 \end{pmatrix}

Therefore, the coordinates of point N are (-2, -6).

If AB : AN = 4:1, then AN = 1/5 - AB

$$\Rightarrow AN =\frac{1}{5} (\vec{OB} - \vec{OA})$$

$$=\frac{1}{5} \begin{pmatrix} -14 -(-6) \\ -16 - 8 \end{pmatrix}$$

$$=\frac{1}{5} \begin{pmatrix} -8 \\ -24 \end{pmatrix}$$

$$=\begin{pmatrix} -\frac{8}{5} \\ -\frac{24}{5} \end{pmatrix}$$

Therefore, the coordinates of point N are (-8/5, -24/5).