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If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
Section 12.2 : Equations of Lines
For problems 1 – 4 give the equation of the line in vector form, parametric form and symmetric form.
- The line through the points \(\left( {7, - 3,1} \right)\) and \(\left( { - 2,1,4} \right)\).
- The line through the point \(\left( {1, - 5,0} \right)\) and parallel to the line given by \(\vec r\left( t \right) = \left\langle {8 - 3t, - 10 + 9t, - 1 - t} \right\rangle \).
- The line through the point \(\left( {1, - 7,14} \right)\) and parallel to the line given by \(x = 6t\), \(y = 9\), \(z = 8 - 16t\).
- 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.
- The line given by \(\vec r\left( t \right) = \left\langle {4 - 7t, - 10 + 5t,21 - 4t} \right\rangle \) and the line given by \(\vec r\left( t \right) = \left\langle { - 2 + 3t,7 + 5t,5 + t} \right\rangle \).
- The line through the points \(\left( {10, - 4,18} \right)\) and \(\left( {5,6, - 7} \right)\) and the line given by \(x = 5 + 3t\), \(y = - 6t\), \(z = 1 + 15t\).
- 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.
- The line passing through the points \(\left( {0, - 9, - 1} \right)\) and \(\left( {1,6, - 3} \right)\) and the line given by \(\vec r\left( t \right) = \left\langle { - 9 - 4t,10 + 6t,1 - 2t} \right\rangle \).
- The line given by \(x = 1 + 6t\), \(y = - 1 - 3t\), \(z = 4 + 12t\) and the line given by \(x = 4 + t\), \(y = - 10 - 8t\), \(z = 3 - 5t\).
- The line given by \(\vec r\left( t \right) = \left\langle {14 + 5t, - 3t,1 + 7t} \right\rangle \) and the line given by \(\vec r\left( t \right) = \left\langle {3 - 3t,5 + 2t, - 2 + 4t} \right\rangle \).
- 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.
- 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.
- Which of the three coordinate planes does the line given by \(x = 16t\), \(y = - 4 - 9t\), \(z = 34\) intersect?