two vertices of right triangle ABC are A(-2,6) and C(7,3).
If the right angle is at vertex A and vertex B is on the x-axis,
identify the coordinates of point B
Let \(\vec{A} = \binom{-2}{6}\)
Let \(\vec{B} = \binom{x}{0}\)
Let \(\vec{C} = \binom{7}{3}\)
\(\begin{array}{rcll} \vec{AC} = \ ? \\ \vec{AC} &=& \vec{C} - \vec{A} \\ \vec{AC} &=& \binom{7}{3} - \binom{-2}{6} \\ \vec{AC} &=& \binom{7+2}{3-6} \\ \vec{AC} &=& \binom{9}{-3} \\ \end{array}\)
\(\begin{array}{rcll} \vec{AB} = \ ? \\ \vec{AB} &=& \vec{B} - \vec{A} \\ \vec{AB} &=& \binom{x}{0} - \binom{-2}{6} \\ \vec{AB} &=& \binom{x+2}{-6} \\ \end{array}\)
\(\triangle ABC\) is a right triangle then \(\vec{AC}\cdot \vec{AB} = 0\)
\(\begin{array}{rcll} \mathbf{ \vec{AC}\cdot \vec{AB} } & \mathbf{=} & \mathbf{0} \\ \binom{9}{-3} \cdot \binom{x+2}{-6} &=& 0 \\ 9\cdot(x+2) + (-3)\cdot (-6) &=& 0 \\ 9x+18+18 &=& 0 \\ 9x+ 36 &=& 0 \quad & \quad :9 \\ x+ 4 &=& 0 \\ \mathbf{x} & \mathbf{=} & \mathbf{-4} \\ \end{array} \)
\(\mathbf{\vec{B} = \binom{-4}{0} } \)