cdb9a9…: “As far as the tall river”

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Find the equation representing the line orthogonal to $\vec {d}_{1}=\vector {1,0,1}$ and $\vec {d}_{2}=\vector {2,0,3}$ that passes through $P=(5,1,9)$ . Express your answer as a vector and as parametric equations. $\text {Vector: } \vecl (t)=\vector {\answer {5},\answer {1},\answer {9}}+t\vector {\answer {0},-1,\answer {0}}$ $\text {Parametric: } x(t)=\answer {5},\ y(t)=\answer {1-t},\ z(t)=\answer {9}$

Let’s first find the direction that the line is heading in.

We are told that the line is orthogonal to $\vec {d}_{1}=\vector {1,0,1}$ and $\vec {d}_{2}=\vector {2,0,3}$ . We can use the cross product to find a vector orthogonal to these two vectors.

$\vector {1,0,1} \cross \vector {2,0,3} = \vector {0,-1,0}$ .

So $\vecl (t) = \mbox {point} + t\vector {0,-1,0}$ .

Since we are told that the line passes through $P$ , we then may conclude $\vecl (t) = \vector {5,1,9} + t\vector {0,-1,0}$ .

Adding together corresponding entries reveals that $x(t)=5$ .

Moreover $y(t)=1-t$ .

Finally $z(t)=9.$