Chapter 9: Problem 26
In Exercises \(25-28,\) convert the point from spherical coordinates to rectangular coordinates. $$ (12,3 \pi / 4, \pi / 9) $$
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Write an equation whose graph consists of the set of points \(P(x, y, z)\) that are twice as far from \(A(0,-1,1)\) as from \(B(1,2,0)\)
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 29 and 30 . find the direction angles of the vector. $$ \mathbf{u}=3 \mathbf{i}+2 \mathbf{j}-2 \mathbf{k} $$
The vertices of a triangle are given. Determine whether the triangle is an acute triangle, an obtuse triangle, or a right triangle. Explain your reasoning. $$ (2,-7,3),(-1,5,8),(4,6,-1) $$
Find the angle \(\theta\) between the vectors. $$ \mathbf{u}=\cos \left(\frac{\pi}{6}\right) \mathbf{i}+\sin \left(\frac{\pi}{6}\right) \mathbf{j}, \quad \mathbf{v}=\cos \left(\frac{3 \pi}{4}\right) \mathbf{i}+\sin \left(\frac{3 \pi}{4}\right) \mathbf{j} $$
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