91Ó°ÊÓ

Find equations of the spheres with center \((2,-3,6)\) that touch (a) the \(x y\) -plane, \((b)\) the \(y z\) -plane, \((c)\) the \(x z\) -plane.

Find an equation of the largest sphere with center \((5,4,9)\) that is contained in the first octant.

Determine whether the given vectors are orthogonal, parallel, or neither. $$ \begin{array}{l}{\text { (a) } \mathbf{u}=\langle- 5,4,-2\rangle, \quad \mathbf{v}=\langle 3,4,-1\rangle} \\ {\text { (b) } \mathbf{u}= 9 \mathbf{i}-6 \mathbf{j}+3 \mathbf{k}, \quad \mathbf{v}=-6 \mathbf{i}+4 \mathbf{j}-2 \mathbf{k}} \\ {\text { (c) } \mathbf{u}=\langle c, c, c\rangle, \quad \mathbf{v}=\langle c, 0,-c\rangle}\end{array} $$

Find a unit vector that has the same direction as the given vector. $$ -5 \mathbf{i}+3 \mathbf{j}-\mathbf{k} $$

Find an equation of the plane. The plane through the point \((5,3,5)\) and with normal vector \(2 \mathbf{i}+\mathbf{j}-\mathbf{k}\)

Use vectors to decide whether the triangle with vertices \(P(1,-3,-2), Q(2,0,-4),\) and \(R(6,-2,-5)\) is right-angled.

Find a unit vector that has the same direction as the given vector. $$ 8 \mathbf{i}-\mathbf{j}+4 \mathbf{k} $$

Find an equation of the plane. The plane through the point \(\left(-1, \frac{1}{2}, 3\right)\) and with normal vector \(\mathbf{i}+4 \mathbf{j}+\mathbf{k}\)

Describe in words the region of \(\mathbb{R}^{3}\) represented by the equation(s) or inequality. $$ x=5 $$

Find the values of \(x\) such that the angle between the vectors \(\langle 2,1,-1\rangle,\) and \(\langle 1, x, 0\rangle\) is \(45^{\circ} .\)