In finite geometry try to prove that 130 1. The number of points is finite. 2. The number of lines is finite. 3. Each line is on the same number \( s \) of points, where \( s>2 \). 4. Each point is on the same number \( t \) of lines, where \( t>2 \). 5. Each pair of distinct points is on at most one line. 6. Each pair of distinct lines is on at most one point. 7. Not all points are on the same line. 8. There exists at least one line.

(a) Find the dot product \( \mathbf{v} \cdot \mathbf{w} \); (b) find the angle between \( \mathbf{v} \) and \( \mathbf{w} \); (c) state whether the vectors are parallel, orthogonal, or neither. \( \mathbf{v}=\sqrt{3} \mathbf{i}+3 \mathbf{j}, \quad \mathbf{w}=-\mathbf{i}+\mathbf{j} \) (a) \( \mathbf{v} \cdot \mathbf{w}=3-\sqrt{3} \) (Simplify your answer. Type an exact answer, using radicals as needed.) (b) The angle between \( \mathbf{v} \) and \( \mathbf{w} \) is \( \square^{\circ} \). (Type your answer in degrees.)

lengths is a right triangle. Justify your answe receive full credit. 14.60 feet, 80 feet, 100 feet

Use the given vectors to find \( \mathbf{v} \cdot \mathbf{w} \) and \( \mathbf{v} \cdot \mathbf{v} \). \[ \begin{array}{l}\mathbf{v}=-7 \mathbf{i}-3 \mathbf{j}, \mathbf{w}=-9 \mathbf{i}-5 \mathbf{j} \\ \mathbf{v} \cdot \mathbf{w}=78 \text { (Simplify your answer.) } \\ \mathbf{v} \cdot \mathbf{v}=\square \text { (Simplify your answer.) }\end{array} \text { \$ } \]

Given \( \mathbf{v}=-9 \mathbf{i} \) and \( \mathbf{w}=-\mathbf{j} \) (a) find the dot product \( \mathbf{v} \cdot \mathbf{w} \) (b) find the angle between \( \mathbf{v} \) and \( \mathbf{w} \); (c) state whether the vectors are parallel, orthogonal, or neither.

Determina la pendiente de la línea que pasa por los puntos \( (-1, -1) \) y \( (2, 5) \).