Given the 3 Vertices A(3,-1) B(7,-4) C(1,-7) of a triangle, calculate the vector equation and the explicit equation of the height corresponding to side BC

To find the vector representing side \overrightarrow{BC} , we subtract the coordinates of point B from the coordinates of point C:

\overrightarrow{BC} = \begin{pmatrix} 7 \ -2 \end{pmatrix} - \begin{pmatrix} 4 \ 5 \end{pmatrix} = \begin{pmatrix} 3 \ -7 \end{pmatrix}

The normal vector to side BC is \begin{pmatrix} 3 \ -7 \end{pmatrix} , which is the direction vector of the height from vertex A.

The vector equation of the height from A to BC is given by:

\vec{r}(t) = \begin{pmatrix} 3 \ -1 \end{pmatrix} + t \begin{pmatrix} 3 \ -7 \end{pmatrix}

And the explicit (Cartesian) equation of the height is:

y = -\frac{3}{7}x - \frac{22}{7}

\boxed{y = -\frac{3}{7}x - \frac{22}{7}}

This line passes through point A(3, -1) and is perpendicular to the line segment BC.