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Chapter 9: Problem 83
Give the standard equation of a sphere of radius \(r\), centered at \(\left(x_{0}, y_{0}, z_{0}\right)\)
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Find the angle between a cube's diagonal and one of its edges.
Find the vector \(z,\) given that \(\mathbf{u}=\langle 1,2,3\rangle\) \(\mathbf{v}=\langle 2,2,-1\rangle,\) and \(\mathbf{w}=\langle 4,0,-4\rangle\) \(2 \mathbf{z}-3 \mathbf{u}=\mathbf{w}\)
Think About It In Exercises \(65-68\), find inequalities that describe the solid, and state the coordinate system used. Position the solid on the coordinate system such that the inequalities are as simple as possible. The solid between the spheres \(x^{2}+y^{2}+z^{2}=4\) and \(x^{2}+y^{2}+z^{2}=9,\) and inside the cone \(z^{2}=x^{2}+y^{2}\)
Let \(\mathbf{u}=\mathbf{i}+\mathbf{j}, \mathbf{v}=\mathbf{j}+\mathbf{k},\) and \(\mathbf{w}=a \mathbf{u}+b \mathbf{v} .\) (a) Sketch \(\mathbf{u}\) and \(\mathbf{v}\). (b) If \(\mathbf{w}=\mathbf{0}\), show that \(a\) and \(b\) must both be zero. (c) Find \(a\) and \(b\) such that \(\mathbf{w}=\mathbf{i}+2 \mathbf{j}+\mathbf{k}\). (d) Show that no choice of \(a\) and \(b\) yields \(\mathbf{w}=\mathbf{i}+2 \mathbf{j}+3 \mathbf{k}\).
In Exercises \(51-56,\) find the vector \(z,\) given that \(\mathbf{u}=\langle 1,2,3\rangle\) \(\mathbf{v}=\langle 2,2,-1\rangle,\) and \(\mathbf{w}=\langle 4,0,-4\rangle\) \(\mathbf{z}=\mathbf{u}-\mathbf{v}\)
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