# How can I find the shortest distance between the point #(0,1,-1)# and the line #(x,y,z) = (2,1,3) + t(3,-1,-2)#?

##### 3 Answers

#### Explanation:

Let

with

The square distance between

substituting

Developing we have

Here

The minimum distance from

This occurs at a point

or

or simplifying

Here

Finally, substituting values we obtain.

We have line:

and point

Let

Now,

And **perpendicular** to

Therefore:

From

Use the direction of the line to find the general equation of the plane.

Use the point to find the specific equation.

Use parametric formulas to find the point on the plane

Use the distance formula.

#### Explanation:

The direction of the given line is the vector

Because the above vector must be a normal vector to the plane that contains the point closest to the given point, we know that its general form is:

Find the specific plane that contains the given point,

The plane that contains the nearest point to the point

The parametric equations of the line are:

Substitute these equations into equation [1] and solve for t:

Using the parametric equations and the value of t, we can find the nearest point:

Use the distance formula to find the distance from,