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Paul
May 6, 2021

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Section 1-2 : 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?