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### Section 12.2 : Equations of Lines

For problems 1 & 2 give the equation of the line in vector form, parametric form and symmetric form.

1. The line through the points $$\left( {2, - 4,1} \right)$$ and $$\left( {0,4, - 10} \right)$$. Solution
2. The line through the point $$\left( { - 7,2,4} \right)$$ and parallel to the line given by $$x = 5 - 8t$$, $$y = 6 + t$$, $$z = - 12t$$. Solution
3. Is the line through the points $$\left( {2,0,9} \right)$$ and $$\left( { - 4,1, - 5} \right)$$ parallel, orthogonal or neither to the line given by $$\vec r\left( t \right) = \left\langle {5,1 - 9t, - 8 - 4t} \right\rangle$$? Solution

For problems 4 & 5 determine the intersection point of the two lines or show that they do not intersect.

1. The line given by $$x = 8 + t$$, $$y = 5 + 6t$$, $$z = 4 - 2t$$ and the line given by $$\vec r\left( t \right) = \left\langle { - 7 + 12t,3 - t,14 + 8t} \right\rangle$$. Solution
2. The line passing through the points $$\left( {1, - 2,13} \right)$$ and $$\left( {2,0, - 5} \right)$$ and the line given by $$\vec r\left( t \right) = \left\langle {2 + 4t, - 1 - t,3} \right\rangle$$. Solution
3. Does the line given by $$x = 9 + 21t$$, $$y = - 7$$, $$z = 12 - 11t$$ intersect the xy-plane? If so, give the point. Solution
4. Does the line given by $$x = 9 + 21t$$, $$y = - 7$$, $$z = 12 - 11t$$ intersect the xz-plane? If so, give the point. Solution