How Do You Calculate the Length of AB When It Is Perpendicular to Line L?

AI Thread Summary
To calculate the length of AB, which is perpendicular to line L, the position vector of point A is given as 30i - 3j - 5k, and line L is defined by the vector 4i - 5j - 3k. The user attempted to express point B on line L in terms of a parameter, leading to the vector AB. They encountered difficulty in solving the perpendicularity condition, AB.L=0, due to having two unknowns. After realizing a missed reference in their textbook, they indicated they could continue with the solution. The discussion highlights the importance of understanding vector relationships and plane equations in 3D geometry.
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1. With respect to origin O, point A has position vector 30i - 3j - 5k. The line L passes through O and is parallel to the vector 4i - 5j -3k. The point B on L is such that AB is perpendicular to L. Find the length of AB. Find the position vector of B.



The Attempt at a Solution


Ok this is what i did.
Let L=y(4i -5j -3k)
Let b=x(4i -5j -3k)
Hence AB = [(4x-30)i (-5x+3)j (-3x+5)k]

Heres where i got quite stuck

Since AB is perpendicular to L. I tried AB.L=0
But I'm not very sure how do i solve this since there are 2 unknowns.

Ahh! Please help me out here! Thanks!
 
Physics news on Phys.org
What is the equation of the plane containing A and having 4i- 5j- 3k as normal vector?

Where does the line L intersect that plane? That is point B.
 
HallsofIvy said:
What is the equation of the plane containing A and having 4i- 5j- 3k as normal vector?

Where does the line L intersect that plane? That is point B.

Ahhh! Thanks! I missed out on the line that was on the next page of the book. :eek: Ok i can take it from here
 

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