91Ó°ÊÓ

Find the magnitude of \(v\). $$ \mathbf{v}=\langle 12,-5\rangle $$

Determine whether the lines intersect, and if so, find the point of intersection and the cosine of the angle of intersection.\(x=2 t-1, y=-4 t+10, z=t\) \(x=-5 s-12, y=3 s+11, z=-2 s-4\)

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

Find the direction cosines of \(\mathbf{u}\) and demonstrate that the sum of the squares of the direction cosines is \(1 .\) $$ \mathbf{u}=5 \mathbf{i}+3 \mathbf{j}-\mathbf{k} $$

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

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. \((5,0,0),(0,2,0),(0,0,-3)\)

Convert the point from rectangular coordinates to spherical coordinates. \((\sqrt{3}, 1,2 \sqrt{3})\)

Find the magnitude of \(v\). $$ \mathbf{v}=6 \hat{\mathbf{i}}-5 \mathbf{j} $$

find the area of the triangle with the given vertices. (Hint: \(\frac{1}{2}\|\mathbf{u} \times \mathbf{v}\|\) is the area of the triangle having \(\mathbf{u}\) and \(\mathbf{v}\) as adjacent sides. $$ (0,0,0),(1,2,3),(-3,0,0) $$

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.) $$ z^{2}=x^{2}+4 y^{2} $$