Chapter 9: Problem 60
In Exercises \(55-60\), convert the rectangular equation to an equation in (a) cylindrical coordinates and (b) spherical coordinates $$ y=4 $$
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In Exercises 77 and \(78,\) use vectors to find the point that lies two-thirds of the way from \(P\) to \(Q\). \(P(4,3,0), \quad Q(1,-3,3)\)
In Exercises 45 and \(46,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \text { If } \mathbf{u} \cdot \mathbf{v}=\mathbf{u} \cdot \mathbf{w} \text { and } \mathbf{u} \neq \mathbf{0}, \text { then } \mathbf{v}=\mathbf{w} $$
In Exercises 69 and \(70,\) find a unit vector \((a)\) in the direction of \(\mathbf{u}\) and \((\mathbf{b})\) in the direction opposite \(\mathbf{u}\) \(\mathbf{u}=\langle 2,-1,2\rangle\)
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 \(\mathbf{w}=\mathbf{i}+2 \mathbf{j}+3 \mathbf{k}\).
Find the vector \(z,\) given that \(\mathbf{u}=\langle 1,2,3\rangle\) \(\mathbf{v}=\langle 2,2,-1\rangle,\) and \(\mathbf{w}=\langle 4,0,-4\rangle\) \(\mathbf{z}=5 \mathbf{u}-3 \mathbf{v}-\frac{1}{2} \mathbf{w}\)
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