Distance Between a Point and a Line Problem: Figure 10-9b shows a line and a point P1

Chapter 10, Problem C.1

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Distance Between a Point and a Line Problem: Figure 10-9b shows a line and a point \(P_{1}\) not on the line. Vector \(\vec v\) is parallel to the line, and d is the perpendicular distance between \(P_{1}\) and the line.

Now suppose that the equation of the line is

\(\vec{r}=\left(5+\frac{6}{11} t\right) \vec{i}+\left(3-\frac{2}{11} t\right) \vec{j}+\left(-1+\frac{9}{11} t\right) \vec{k}\)

and the outside point is \(P_{1}(4,7,6)\).

a. By trigonometry, . Multiply the right-hand side of this equation by 1 in the form . Take advantage of the fact that the numerator on the right now equals to find d without first finding .

b. Ladder Problem 2: Figure 10-9c shows a 25-ft ladder leaning against a wall to reach a high window. To miss the flower bed, the ladder is moved over so that its left foot is at (7, 9, 0). The top of the ladder is 24 ft up the wall. Find a vector equation of the line along the left side of the ladder. Given that the rungs on the ladder are 1 ft apart, use the equation to find the rung that is closest to the upper left corner of the window, at the point (0, 5, 18). Find the perpendicular distance from the left side of the ladder to (0, 5, 18), taking advantage of the results of part a,

\(d=\frac{\left|\overrightarrow{P P_{1}} \times \vec{v}\right|}{|\vec{v}|}\)

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