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

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SUMMARY

The discussion focuses on calculating the length of segment AB, which is perpendicular to line L defined by the vector 4i - 5j - 3k and passes through the origin O. The position vector of point A is given as 30i - 3j - 5k. The solution involves setting up the equation AB.L = 0 to ensure perpendicularity, leading to the determination of point B on line L. The user ultimately resolves the problem by referencing additional material from their textbook.

PREREQUISITES
  • Understanding of vector algebra and position vectors
  • Knowledge of the concept of perpendicular vectors
  • Familiarity with parametric equations of lines in three-dimensional space
  • Basic comprehension of planes defined by normal vectors
NEXT STEPS
  • Study the properties of perpendicular vectors in vector calculus
  • Learn how to derive equations of planes given a point and a normal vector
  • Explore the concept of parametric equations for lines in 3D geometry
  • Practice solving problems involving intersections of lines and planes
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Students and educators in mathematics, particularly those studying vector calculus and three-dimensional geometry, as well as anyone looking to enhance their problem-solving skills in spatial reasoning.

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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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