Finding the coordinates of a point on a line: Vectors

[lunds002]

Consider the points A (1,3,-17) and B (6,-7,8) which lie on the line L.

a) find an equation of line L in parametric form.

I found vector AB=(5,-10,25), and so I found the equation to be x=1-5t, y=3+10t, z=-17-25t

b) The point P is on line L such that vector OP is perpendicular to L. Find the coordinates of point P.

I know that OP is perpendicular to the line L if the dot product of vector AB and OP equals zero, but I'm not sure if that will help me find a solution to part b. Help?

 

[zhermes]

I know that OP is perpendicular to the line L if the dot product of vector AB and OP equals zero, but I'm not sure if that will help me find a solution to part b.

If the vector is perpendicular to AB, what is its direction?
Using that, you can construct a dot product with the point P as a variable, then solve.

 

[lunds002]

I'm unsure of how to find the direction vector..

 

[Mark44]

If the vector is perpendicular to AB, what is its direction?
Using that, you can construct a dot product with the point P as a variable, then solve.

I'm unsure of how to find the direction vector..

Can you find a vector OP, from the origin to an arbitrary point on your line? Since OP is perpendicular to the line, OP $\cdot$ AB = 0.

 

[lunds002]

No.. I struggle with vectors so I don't really know how to do that.

 

[Mark44]

Any point on your line has coordinates <1 - 5t, 3 + 10t, -17 - 25t>, so this is the same as the vector OP.

Set the dot product of this vector and AB to zero, and solve for t. That will give you the point P on your line such that OP is perpendicular to AB.

 

[lunds002]

Ohh that makes sense, thanks so much! I got the answer now.