91Ó°ÊÓ

Find parametric equations for the line that passes through the points \(P\) and \(Q .\) $$ P(12,16,18), \quad Q(12,-6,0) $$

Find the magnitude of the given vector. $$ \langle 1,-6,2 \sqrt{2}\rangle $$

\(11-14\) . Find an equation of a sphere with the given radius \(r\) and center \(C .\) $$ r=\sqrt{11} ; \quad \quad \quad(-10,0,1) $$

Find (a) \(\mathbf{u} \cdot \mathbf{v}\) and (b) the angle between \(\mathbf{u}\) and \(\mathbf{v}\) to the nearest degree. $$ \mathbf{u}=3 \mathbf{i}+4 \mathbf{j}, \quad \mathbf{v}=-2 \mathbf{i}-\mathbf{j} $$

The lengths of two vectors \(\mathbf{a}\) and \(\mathbf{b}\) and the angle \(\theta\) between them are given. Find the length of their cross product, \(|\mathbf{a} \times \mathbf{b}|\). $$ |\mathbf{a}|=4, \quad|\mathbf{b}|=5, \quad \theta=30^{\circ} $$

\(15-18=\) Show that the equation represents a sphere, and find its center and radius. $$ x^{2}+y^{2}+z^{2}-10 x+2 y+8 z=9 $$

Determine whether the given vectors are perpendicular. $$ \mathbf{u}=\langle 6,4\rangle, \quad \mathbf{v}=\langle- 2,3\rangle $$

The lengths of two vectors \(\mathbf{a}\) and \(\mathbf{b}\) and the angle \(\theta\) between them are given. Find the length of their cross product, \(|\mathbf{a} \times \mathbf{b}|\). $$ |\mathbf{a}|=10, \quad|\mathbf{b}|=10, \quad \theta=90^{\circ} $$

Find the vectors \(\mathbf{u}+\mathbf{v}, \mathbf{u}-\mathbf{v},\) and \(3 \mathbf{u}-\frac{1}{2} \mathbf{v}\) $$ \mathbf{u}=\langle 2,-7,3\rangle, \mathbf{v}=\langle 0,4,-1\rangle $$

\(9-18\) . Express the vector with initial point \(P\) and terminal point \(Q\) in component form. $$ P(5,3), \quad Q(1,0) $$