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Chapter 11: Problem 21
Sketch the plane parallel to the \(x y\) -plane through (2,4,2) and find its equation.
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Prove or disprove For fixed values of \(a, b, c,\) and \(d,\) the value of proj \(_{(k a, k b)}\langle c, d\rangle\) is constant for all nonzero values of \(k,\) for \(\langle a, b\rangle \neq\langle 0,0\rangle\).
Proof of Product Rule By expressing \(\mathbf{u}\) in terms of its components, prove that $$\frac{d}{d t}(f(t) \mathbf{u}(t))=f^{\prime}(t) \mathbf{u}(t)+f(t) \mathbf{u}^{\prime}(t)$$
An ant walks due east at a constant speed of \(2 \mathrm{mi} / \mathrm{hr}\) on a sheet of paper that rests on a table. Suddenly, the sheet of paper starts moving southeast at \(\sqrt{2} \mathrm{mi} / \mathrm{hr} .\) Describe the motion of the ant relative to the table.
Jack pulls east on a rope attached to a camel with a force of 40 ib. Jill pulls north on a rope attached to the same camel with a force of 30 Ib. What is the magnitude and direction of the force on the camel? Assume the vectors lie in a horizontal plane.
Compute the following derivatives. $$\frac{d}{d t}\left(t^{2}(\mathbf{i}+2 \mathbf{j}-2 t \mathbf{k}) \cdot\left(e^{t} \mathbf{i}+2 e^{t} \mathbf{j}-3 e^{-t} \mathbf{k}\right)\right)$$
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