5.1 2D Vectors (applications)

Use the parametric form to represent the line which passes through two points [math]A(1,7)[/math] and [math]B(9,-1)[/math].

Let [math]A(-4,3)[/math] and [math]B(9,-1)[/math] be two points. [math]C[/math] is a third point such that [math]AC:AB=2:5[/math] and [math]A, B[/math], and [math]C[/math] are collinear. If [math]C[/math] is in the second quadrant, find the coordinates of [math]C[/math].

Let [math]A(1,6), B(-4,-1)[/math], and [math]C[/math] be three points. If [math]P(0,-2)[/math] is another point that satisfies $$\overrightarrow{PA}+\overrightarrow{PB}+\overrightarrow{PC}=\overrightarrow{0}$$
what are the coordinates of [math]C[/math]?

(Calculator) Consider a unit circle with center [math]O(0,0)[/math]. Two points [math]A[/math], [math]B[/math] are on the unit circle and [math]\angle AOB=40^\circ[/math]. Let [math]P[/math] be a point satisfying [math]\overrightarrow{OP}=t\overrightarrow{OA}+s\overrightarrow{OB}[/math] where [math]t[/math] and [math]s[/math] are nonnegative numbers and [math]s+t\leq1[/math], then all possible [math]P[/math]s can form a region. What is the area of the region?

Consider a triangle [math]ABC[/math]. Point [math]D[/math] is on [math]\overline{BC}[/math]. [math]\overline{AD}[/math] is the angle bisector of [math]\angle A[/math]. Suppose [math]AB=12, AC=6[/math] and [math]BC=9[/math].

(a) Find [math]BD[/math] and [math]DC[/math].

(b) If [math]\overrightarrow{AD}=p\overrightarrow{AB}+q\overrightarrow{AC}[/math] where [math]p[/math] and [math]q[/math] are constants, what are [math]p[/math] and [math]q[/math]?

(c) Evaluate [math]\cos B[/math].

(d) If [math]AD=\sqrt{a}[/math] where [math]a[/math] is an integer, what is the value of [math]a[/math]?

(e) Let [math]I[/math] be the incenter of [math]ABC[/math]. If [math]\overrightarrow{AI}=x\overrightarrow{AB}+y\overrightarrow{AC}[/math]. where [math]x[/math] and [math]y[/math] are constants, what are [math]x[/math] and [math]y[/math]?

Find the intersection of lines [math]L_1: \begin{cases}x=1+2t\\y=4-3t\end{cases}[/math] and [math]L_2: \begin{cases}x=-7-5t\\y=5+2t\end{cases}[/math].

Consdier triangle [math]ABC[/math] with vertices [math]A(-1,-5), B(-4,6)[/math], and [math]C(3,-1)[/math]. [math]D[/math] is on [math]\overline{BC}[/math] and [math]3DB=4DC[/math]. Find the coordinates of [math]D[/math] and the area of [math]ABD[/math].