Chapter 23: Chapter 23

Problem 589

Give an example to show that two non-parallel lines in \(\mathrm{R}^{3}\) need not intersect.

Short Answer

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An example of two non-parallel lines in \(\mathrm{R}^{3}\) that do not intersect are \( L_{1}: (x, y, z) = (1, 2, 3) + t(1, -1, 0) \) and \( L_{2}: (x, y, z) = (2, 4, 1) + s(1, 1, -1) \). The direction vectors for these lines are \( \vec{a} = (1, -1, 0) \) and \( \vec{b} = (1, 1, -1) \), which are not proportional, confirming that the lines are non-parallel. Solving for parameters t and s to check for intersection points, we find that they do not have a unique intersection point, indicating that the lines do not intersect. This example shows that in \(\mathrm{R}^{3}\), two non-parallel lines do not necessarily intersect.

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Step 1: Choosing two non-parallel lines in \(\mathrm{R}^{3}\)

To choose two non-parallel lines, we need to make sure that their direction vectors are not proportional, meaning that there isn't a single constant that can be multiplied by one direction vector to obtain the other. Let's choose two lines in \(\mathrm{R}^{3}\) with the following parametric equations: \( L_{1}: (x, y, z) = (1, 2, 3) + t(1, -1, 0) \) \( L_{2}: (x, y, z) = (2, 4, 1) + s(1, 1, -1) \)

Step 2: Check if the lines are non-parallel

To check if the lines are non-parallel, we will see if their direction vectors are not proportional. The direction vectors for the lines are: \( \vec{a} = (1, -1, 0) \) for line \( L_{1} \) and \( \vec{b} = (1, 1, -1) \) for line \( L_{2} \). No constant can be multiplied by the individual components of either vector to obtain the other. Therefore, since these direction vectors are not proportional, the lines are non-parallel.

Step 3: Check if the lines intersect

To check if the lines intersect, we need to see if there exists a point that lies on both lines. Set the equations of the two lines equal to each other and try to solve for parameters t and s: \( (1, 2, 3) + t(1, -1, 0) = (2, 4, 1) + s(1, 1, -1) \) This gives us the following system of equations: 1. \( 1 + t = 2 + s \) 2. \( 2 - t = 4 + s \) 3. \( 3 = 1 - s \) From equation 3, we can see that \( s = -2 \). Plugging this value into equations 1 and 2: 1. \( 1 + t = 2 - 2 \Rightarrow t = -1 \) 2. \( 2 - t = 4 - 2 \Rightarrow t = 0 \) As we got two different values of t corresponding to the same value of s (i.e., the t value was not unique), we can conclude there is no intersection point, and hence, no point where the two lines overlap. We have shown that the two lines \( L_{1} \) and \( L_{2} \) are non-parallel, and they do not intersect. This example demonstrates that two non-parallel lines in \(\mathrm{R}^{3}\) do not necessarily intersect.

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Chapter 23: Chapter 23

Give an example to show that two non-parallel lines in \(\mathrm{R}^{3}\) need not intersect.

Short Answer

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Step 1: Choosing two non-parallel lines in \(\mathrm{R}^{3}\)

Step 2: Check if the lines are non-parallel

Step 3: Check if the lines intersect

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Chapter 1

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Chapter 3

Chapter 4

Chapter 5

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Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 13

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Chapter 15

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Chapter 18

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