# Find parametric equations for the line through the point (0,1,2)

Find parametric equations for the line through the point (0,1,2) that is perpendicular to the line x=1+t, y=1-t, z=2t and intersects this line.

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Novembruuo

Let us name (0,1,2) as the point A
Point B is on the given line such that AB is perpendicular to the given line
Vector along the given line $$\displaystyle{r}_{{1}}={<}{1},-{1},{2}{>}$$
Vector along the given line A and B is
$$\displaystyle{r}_{{2}}={<}{1}+{t},{1}-{t},{2}{t}\succ{<}{0},{1},{2}\ge{<}{1}+{t},-{t},{2}{t}-{2}{>}$$
$$\displaystyle{r}_{{1}}$$ and $$\displaystyle{r}_{{2}}$$ must be perpendicular, hence their dot product must be 0
$$\displaystyle{<}{1},-{1},{2}{>}\cdot{<}{1}+{t},-{t},{2}{t}-{2}\ge{0}$$
$$\displaystyle{\left({1}+{t}\right)}+{t}+{\left({4}{t}-{4}\right)}={0}$$
$$\displaystyle{6}{t}-{3}={0}$$
$$\displaystyle{6}{t}={3}$$
$$\displaystyle{t}={0.5}$$
The required line passes through (0,1,2)
Required line is parallel to $$\displaystyle{r}_{{2}}={<}{1}+{0.5},-{0.5},{2}\cdot{0.5}-{2}\ge{<}{1.5},-{0.5},-{1}{>}$$
Recall that: Vector equation of a line passing a point with position vector a and parallel to the vector b is $$\displaystyle{r}{\left({t}\right)}={a}+{t}{b}$$
Therefore equation of the required line is
$$\displaystyle{r}{\left({t}\right)}={<}{0},{1},{2}{>}+{t}{<}{1.5},-{0.5},-{1}{>}$$
$$\displaystyle{r}{\left({t}\right)}={<}{1.5}{t},{1}-{0.5}{t},{2}-{t}{>}$$
Parametric equations are
$$\displaystyle{x}={1.5}{t},\ {y}={1}-{0.5}{t},\ {z}={2}-{t}$$
Result: $$\displaystyle{x}={1.5}{t},\ {y}={1}-{0.5}{t},\ {z}={2}-{t}$$