91Ó°ÊÓ

The vector \(\mathbf{v}\) and its initial point are given. Find the terminal point. $$ \mathbf{v}=\langle 4,-9\rangle ; \text { Initial point: }(3,2) $$

The vertices of a triangle are given. Determine whether the triangle is an acute triangle, an obtuse triangle, or a right triangle. Explain your reasoning. $$ (2,-7,3),(-1,5,8),(4,6,-1) $$

Find the direction cosines of \(\mathbf{u}\) and demonstrate that the sum of the squares of the direction cosines is \(1 .\) $$ \mathbf{u}=\mathbf{i}+2 \mathbf{j}+2 \mathbf{k} $$

Verify that the points are the vertices of a parallelogram, and find its area. $$ (1,1,1),(2,3,4),(6,5,2),(7,7,5) $$

Convert the point from rectangular coordinates to spherical coordinates. \((-2,2 \sqrt{3}, 4)\)

Find the lengths of the sides of the triangle with the indicated vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither. \((1,-3,-2),(5,-1,2),(-1,1,2)\)

Find the magnitude of \(v\). $$ \mathbf{v}=\langle 4,3\rangle $$

Use a computer algebra system to graph the surface. (Hint: It may be necessary to solve for \(z\) and acquire two equations to graph the surface.) $$ z=2 \sin x $$

Determine whether the lines intersect, and if so, find the point of intersection and the cosine of the angle of intersection.\(x=2 t+3, y=5 t-2, z=-t+1\) \(x=-2 s+7, y=s+8, z=2 s-1\)

Use a computer algebra system to graph the surface. (Hint: It may be necessary to solve for \(z\) and acquire two equations to graph the surface.) $$ z=x^{2}+0.5 y^{2} $$