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Section 12.2 : Equations of Lines

For problems 1 – 4 give the equation of the line in vector form, parametric form and symmetric form.

1. The line through the points \(\left( {7, - 3,1} \right)\) and \(\left( { - 2,1,4} \right)\).
2. The line through the point \(\left( {1, - 5,0} \right)\) and parallel to the line given by \(\vec r\left( t \right) = \left\langle {8 - 3t, - 10 + 9t, - 1 - t} \right\rangle \).
3. The line through the point \(\left( {1, - 7,14} \right)\) and parallel to the line given by \(x = 6t\), \(y = 9\), \(z = 8 - 16t\).
4. The line through the point \(\left( { - 7,2,4} \right)\) and orthogonal to both \(\vec v = \left\langle {0, - 9,1} \right\rangle \) and \(\vec w = 3\vec i + \vec j - 4\vec k\).

For problems 5 – 7 determine if the two lines are parallel, orthogonal or neither.

1. The line given by \(\vec r\left( t \right) = \left\langle {4 - 7t, - 10 + 5t,21 - 4t} \right\rangle \) and the line given by \(\vec r\left( t \right) = \left\langle { - 2 + 3t,7 + 5t,5 + t} \right\rangle \).
2. The line through the points \(\left( {10, - 4,18} \right)\) and \(\left( {5,6, - 7} \right)\) and the line given by \(x = 5 + 3t\), \(y = - 6t\), \(z = 1 + 15t\).
3. The line given by \(x = 29\), \(y = - 3 - 6t\), \(z = 12 - t\) and the line given by \(\vec r\left( t \right) = \left\langle {12 - 14t,2 + 7t, - 10 + 3t} \right\rangle \).

For problems 8 10 determine the intersection point of the two lines or show that they do not intersect.

1. The line passing through the points \(\left( {0, - 9, - 1} \right)\) and \(\left( {1,6, - 3} \right)\) and the line given by \(\vec r\left( t \right) = \left\langle { - 9 - 4t,10 + 6t,1 - 2t} \right\rangle \).
2. The line given by \(x = 1 + 6t\), \(y = - 1 - 3t\), \(z = 4 + 12t\) and the line given by \(x = 4 + t\), \(y = - 10 - 8t\), \(z = 3 - 5t\).
3. The line given by \(\vec r\left( t \right) = \left\langle {14 + 5t, - 3t,1 + 7t} \right\rangle \) and the line given by \(\vec r\left( t \right) = \left\langle {3 - 3t,5 + 2t, - 2 + 4t} \right\rangle \).
4. Does the line passing through \(\left( { - 5,4, - 1} \right)\) and \(\left( { - 3, - 5,0} \right)\) intersect the yz-plane? If so, give the point.
5. Does the line given by \(\vec r\left( t \right) = \left\langle {6 + t, - 8 + 14t,4t} \right\rangle \) intersect the xz-plane? If so, give the point.
6. Which of the three coordinate planes does the line given by \(x = 16t\), \(y = - 4 - 9t\), \(z = 34\) intersect?