Question

How do you find a vector equation for the line through the point (5, 1, 4) and perpendicular to the plane x - 2y + z = 1?

Answer 1

`vec r = ((5),(1),(4)) + lamda ((1),(-2),(1))`

Explanation:

If we form a vector with the coefficients of the plane equation this will be Normal (ie perpendiculr) to the plane.

So the plane equation is `x - 2y + z = 1`

Hence, the vector `vec n=((1),(-2),(1))` is Normal to that plane

So we want our line to pass through `(5,1,4)` and to be in the direction of `vec n`, so it will have an equation of the form:

`vec r = vec a + lamda vecn`

`:. vec r = ((5),(1),(4)) + lamda ((1),(-2),(1))` is the req'd equation