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Chapter 12: Problem 38
Calculate the work done in the following situations. A stroller is pushed \(20 \mathrm{m}\) with a constant force of \(10 \mathrm{N}\) at an angle of \(15^{\circ}\) below the horizontal.
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Prove the following identities. Assume that \(\mathbf{u}, \mathbf{v}, \mathbf{w}\) and \(\mathbf{x}\) are nonzero vectors in \(\mathbb{R}^{3}\). $$(\mathbf{u} \times \mathbf{v}) \cdot(\mathbf{w} \times \mathbf{x})=(\mathbf{u} \cdot \mathbf{w})(\mathbf{v} \cdot \mathbf{x})-(\mathbf{u} \cdot \mathbf{x})(\mathbf{v} \cdot \mathbf{w})$$
A pair of lines in \(\mathbb{R}^{3}\) are said to be skew if they are neither parallel nor intersecting. Determine whether the following pairs of lines are parallel, intersecting, or skew. If the lines intersect. determine the point(s) of intersection. $$\begin{aligned} &\mathbf{r}(t)=\langle 1+6 t, 3-7 t, 2+t\rangle;\\\ &\mathbf{R}(s)=\langle 10+3 s, 6+s, 14+4 s\rangle \end{aligned}$$
Let \(\mathbf{u}=\left\langle u_{1}, u_{2}, u_{3}\right\rangle\) \(\mathbf{v}=\left\langle v_{1}, v_{2}, v_{3}\right\rangle\), and \(\mathbf{w}=\) \(\left\langle w_{1}, w_{2}, w_{3}\right\rangle\). Let \(c\) be a scalar. Prove the following vector properties. \(c(\mathbf{u} \cdot \mathbf{v})=(c \mathbf{u}) \cdot \mathbf{v}=\mathbf{u} \cdot(c \mathbf{v})\)
Determine the equation of the line that is perpendicular to the lines \(\mathbf{r}(t)=\langle 4 t, 1+2 t, 3 t\rangle\) and \(\mathbf{R}(s)=\langle-1+s,-7+2 s,-12+3 s\rangle\) and passes through the point of intersection of the lines \(\mathbf{r}\) and \(\mathbf{R}\).
Consider the lines $$\begin{aligned} \mathbf{r}(t) &=\langle 2+2 t, 8+t, 10+3 t\rangle \text { and } \\ \mathbf{R}(s) &=\langle 6+s, 10-2 s, 16-s\rangle. \end{aligned}$$ a. Determine whether the lines intersect (have a common point) and if so, find the coordinates of that point. b. If \(\mathbf{r}\) and \(\mathbf{R}\) describe the paths of two particles, do the particles collide? Assume \(t \geq 0\) and \(s \approx 0\) measure time in seconds, and that motion starts at \(s=t=0\).
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