1.
In triangle ABC, S is a point on side BC such that BS:SC = 1:2,
and T is a point on side AC such that AT:TC = 4:3.
Let U be the intersection of AS and BT.
We can write \( \vec{T} = w \vec{A} + x \vec{C}\), \(\vec{S} = y \vec{B} + z \vec{C}\) for some real values of w, x, y, and z.
Find w, x, y, and z.
\(\begin{array}{|rcll|} \hline \vec{T} &=& \vec{A}+ \dfrac{4}{7}\left(\vec{C}-\vec{A} \right) \\ \vec{T} &=& \vec{A}+ \dfrac{4}{7}\vec{C} -\dfrac{4}{7}\vec{A} \\ \vec{T} &=& \dfrac{3}{7}\vec{A}+ \dfrac{4}{7}\vec{C} \quad &|\quad \vec{T} = w \vec{A} + x \vec{C}\\ \hline \mathbf{w}&=& \mathbf{\dfrac{3}{7}} \\ \mathbf{x}&=& \mathbf{\dfrac{4}{7}} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline \vec{S} &=& \vec{C}+ \dfrac{2}{3}\left(\vec{B}-\vec{C} \right) \\ \vec{S} &=& \vec{C}+ \dfrac{2}{3}\vec{B} -\dfrac{2}{3}\vec{C} \\ \vec{S} &=& \dfrac{2}{3}\vec{B}+ \dfrac{1}{3}\vec{C} \quad &|\quad \vec{S} = y \vec{B} + z \vec{C}\\ \hline \mathbf{y}&=& \mathbf{\dfrac{2}{3}} \\ \mathbf{z}&=& \mathbf{\dfrac{1}{3}} \\ \hline \end{array}\)