Chapter 9: Problem 29
In Exercises \(29-32,\) find the standard equation of the sphere. Center: (0,2,5) Radius: 2
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In Exercises 75 and \(76,\) sketch the vector \(v\) and write its component form. \(\mathbf{v}\) lies in the \(y z\) -plane, has magnitude 2 , and makes an angle of \(30^{\circ}\) with the positive \(y\) -axis.
In Exercises 57-60, determine which of the vectors is (are) parallel to \(\mathrm{z}\). Use a graphing utility to confirm your results. \(\mathrm{z}=\langle 3,2,-5\rangle\) (a) \langle-6,-4,10\rangle (b) \(\left\langle 2, \frac{4}{3},-\frac{10}{3}\right\rangle\) (c) \langle 6,4,10\rangle (d) \langle 1,-4,2\rangle
Find the angle \(\theta\) between the vectors. $$ \begin{array}{l} \mathbf{u}=3 \mathbf{i}+4 \mathbf{j} \\ \mathbf{v}=-2 \mathbf{j}+3 \mathbf{k} \end{array} $$
Find the component of \(u\) that is orthogonal to \(\mathbf{v},\) given \(\mathbf{w}_{\mathbf{1}}=\operatorname{proj}_{\mathbf{v}} \mathbf{u}\). $$ \mathbf{u}=\langle 8,2,0\rangle, \quad \mathbf{v}=\langle 2,1,-1\rangle, \quad \operatorname{proj}_{\mathbf{v}} \mathbf{u}=\langle 6,3,-3\rangle $$
Give the formula for the distance between the points \(\left(x_{1}, y_{1}, z_{1}\right)\) and \(\left(x_{2}, y_{2}, z_{2}\right)\)
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