Chapter 9: Problem 28
Find the coordinates of the midpoint of the line segment joining the points. (4,0,-6),(8,8,20)
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In Exercises 57-60, determine which of the vectors is (are) parallel to \(\mathrm{z}\). Use a graphing utility to confirm your results. \(\mathrm{z}=\langle 3,2,-5\rangle\) (a) \langle-6,-4,10\rangle (b) \(\left\langle 2, \frac{4}{3},-\frac{10}{3}\right\rangle\) (c) \langle 6,4,10\rangle (d) \langle 1,-4,2\rangle
A toy wagon is pulled by exerting a force of 25 pounds on a handle that makes a \(20^{\circ}\) angle with the horizontal. Find the work done in pulling the wagon 50 feet.
The vector \(\mathbf{u}=\langle 3240,1450,2235\rangle\) gives the numbers of hamburgers, chicken sandwiches, and cheeseburgers, respectively, sold at a fast-food restaurant in one week. The vector \(\mathbf{v}=\langle 1.35,2.65,1.85\rangle\) gives the prices (in dollars) per unit for the three food items. Find the dot product \(\mathbf{u} \cdot \mathbf{v},\) and explain what information it gives.
Find \((\mathbf{a}) \mathbf{u} \cdot \mathbf{v},(\mathbf{b}) \mathbf{u} \cdot \mathbf{u},(\mathbf{c})\|\mathbf{u}\|^{2},(\mathbf{d})(\mathbf{u} \cdot \mathbf{v}) \mathbf{v}\) and \((e) u \cdot(2 v)\). $$ \mathbf{u}=\mathbf{i}, \quad \mathbf{v}=\mathbf{i} $$
In Exercises 31 and 32 , find the component of \(u\) that is orthogonal to \(\mathbf{v},\) given \(\mathbf{w}_{\mathbf{1}}=\operatorname{proj}_{\mathbf{v}} \mathbf{u}\). $$ \mathbf{u}=\langle 6,7\rangle, \quad \mathbf{v}=\langle 1,4\rangle, \quad \operatorname{proj}_{\mathbf{v}} \mathbf{u}=\langle 2,8\rangle $$
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