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Chapter 13: Problem 27

Line segments Find parametric equations for the line segment joining the first point to the second point. $$(0,0,0) \text { and }(1,2,3)$$

Short Answer

Expert verified
Answer: The parametric equations for the line segment are \(x=1t\), \(y=2t\), \(z=3t\), where \(0\leq t \leq1\).

Step by step solution

01

Find the direction vector of the line segment

The direction vector, \(\vec{d}\), of the line segment between points A(0,0,0) and B(1,2,3) is obtained by subtracting the position vector of A from the position vector of B. That is: \(\vec{d} = \vec{B} - \vec{A} = (1-0, 2-0, 3-0) = (1, 2, 3)\)
02

Choose a point as the initial point

We can choose either (0,0,0) or (1,2,3) as the initial point. Here, we choose A(0,0,0).
03

Set up parametric equations using the initial point and direction vector

Having the initial point A(0,0,0) and direction vector \(\vec{d} = (1, 2, 3)\), the parametric equations of the line segment are given by: \(x=0+1t\), \(y=0+2t\), \(z=0+3t\), where \(0\leq t \leq1\). So, the final parametric equations for the line segment joining the points (0,0,0) and (1,2,3) are: \(x=1t\), \(y=2t\), \(z=3t\), where \(0\leq t \leq1\).

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