Distance Between Origin and Lin

[TrueStar]

Homework Statement

Find the distance from the origin to the line
x=1+t
y=2-t
z=-1+2t

Homework Equations

Possibly d=|aXb| / |a|

The Attempt at a Solution

I think I need to use the formula provided above, but I'm not sure how. A different distance formula was covered in class, but I think there are different formulas to use based on what you're trying to find the distance between.

There is an example in the textbook like this that asks to use this formula, but it gives out three points and tells assigns vectors for a and b.

I'm not sure how to approach this.

 

[HallsofIvy]

Don't just memorize formulas without learning when and how they apply.

Find the plane perpendicular to the given line containin (0, 0, 0). The shortest distance from the origin to the line must be along a line in that plane. Where does the line cross that plane?

 

[TrueStar]

I tried to find an equation of a plane with the parametric equation given to me. I came up with:

x-y+2z=0 - I plugged in the origin points, which makes d=0.

Using the origin as a point, I applied the distance formula (the absolute value of the equation of a plane with the point plugged in divided by the square root of the sum of the normal vector squared), but I'm getting zero for an answer.

I don't know what I'm doing wrong. Since the origin is a 0,0,0, the equation ofthe line will always cancel out.

 

[ehild]

The distance of a point P from a line is defined as the minimum of the distance of any point on the line from the given point P. Write the distance from the origin of a point (x,y,z) on the line in terms of t and find t where it is minimal.

ehild

 

[HallsofIvy]

I tried to find an equation of a plane with the parametric equation given to me. I came up with:

x-y+2z=0 - I plugged in the origin points, which makes d=0.

Using the origin as a point, I applied the distance formula (the absolute value of the equation of a plane with the point plugged in divided by the square root of the sum of the normal vector squared), but I'm getting zero for an answer.

I don't know what I'm doing wrong. Since the origin is a 0,0,0, the equation ofthe line will always cancel out.

I said before, "Where does the line cross that plane?"

 

[TrueStar]

I think I got the equation of a plane wrong...I think it should be -x-y-3z=0

I used this and the origin points to put into a vector equation:

(0,0,0) + t(-1,-1,-3) - This gives me position vectors of

x=0-t
y=0-t
z=0-3t

I plug this into my equation of a plane which was -x-y-3z=0 so that it looks like:

-1(0-t)-1(0-t)-3(0-3t)=0

This mean t=0. I use this to plug back into the three equations above and I still get (0,0,0). I don't feel this is correct.

 

[ehild]

I think I got the equation of a plane wrong...I think it should be -x-y-3z=0

it should be r˙t=0 where t is the direction vector of the line, (1,-1,2), so x-y+2z=0.
Plug in x=1+t, y=2-t and z=-1+2t, and find t.

ehild

 

[TrueStar]

OK, I think I see what I did wrong with the equation of a line. I think the multiples of t in the parametric equations are the scalers for the equation of a line. I may not have my terminology right here.

So I plugged in the parametric equations for x, y, and z and solved for t. It was 1/2. I understand that this would be a minimum distance.

The points are (3/2, 3/2, 0) and the actual distance is the sqrt of 4.5.

Hopefully this is the correct answer.