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Chapter 12: Problem 26
Find an equation of the line segment joining the first point to the second point. $$(1,0,1) \text { and }(0,-2,1)$$
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Two sides of a parallelogram are formed by the vectors \(\mathbf{u}\) and \(\mathbf{v}\). Prove that the diagonals of the parallelogram are \(\mathbf{u}+\mathbf{v}\) and \(\mathbf{u}-\mathbf{v}\)
Prove that for integers \(m\) and \(n\), the curve $$\mathbf{r}(t)=\langle a \sin m t \cos n t, b \sin m t \sin n t, c \cos m t\rangle$$ lies on the surface of a sphere provided \(a^{2}+b^{2}=c^{2}\).
Consider the parallelogram with adjacent sides \(\mathbf{u}\) and \(\mathbf{v}\). a. Show that the diagonals of the parallelogram are \(\mathbf{u}+\mathbf{v}\) and \(\mathbf{u}-\mathbf{v}\). b. Prove that the diagonals have the same length if and only if \(\mathbf{u} \cdot \mathbf{v}=0\). c. Show that the sum of the squares of the lengths of the diagonals equals the sum of the squares of the lengths of the sides.
A race Two people travel from \(P(4,0)\) to \(Q(-4,0)\) along the paths given by $$ \begin{aligned} \mathbf{r}(t) &=(4 \cos (\pi t / 8), 4 \sin (\pi t / 8)\rangle \text { and } \\\ \mathbf{R}(t) &=\left(4-t,(4-t)^{2}-16\right) \end{aligned} $$ a. Graph both paths between \(P\) and \(Q\) b. Graph the speeds of both people between \(P\) and \(Q\) c. Who arrives at \(Q\) first?
Consider the curve described by the vector function \(\mathbf{r}(t)=\left(50 e^{-t} \cos t\right) \mathbf{i}+\left(50 e^{-t} \sin t\right) \mathbf{j}+\left(5-5 e^{-t}\right) \mathbf{k},\) for \(t \geq 0\). a. What is the initial point of the path corresponding to \(\mathbf{r}(0) ?\) b. What is \(\lim _{t \rightarrow \infty} \mathbf{r}(t) ?\) c. Sketch the curve. d. Eliminate the parameter \(t\) to show that \(z=5-r / 10\), where \(r^{2}=x^{2}+y^{2}\).
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