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# Advanced Math/lines in space : vector equations

Question
find the vector equation of the straight line that passes through
P( -4,1,2) and Q ( 2,7,6 ) in which the parameter u takes value 0 at Q, the midpoint of PR, and value 1 at R. Calculate the length of QR, and hence find the vector equation of this line using the parameter v where v=0 at Q and v measures length along the line

The eqn for a line can be written L = R + tV where P is some point and V is a vector. For this case, V will be parallel to the vector between P and Q: V = Q-P = (2,7,6)-(-4,1,2) = (6,6,4).

We want the parameter t (or u or v in your example) to have L = Q when t = 0. We also know that Q is the mid-point between P and R and that L = R when t = 1, which also means L = P when t = -1. The eqn L = Q + t(Q-P) does this.

In the future, please show your work as far as you can and/or ask a specific question rather than just sending a homework problem.

Thanks, Randy

Volunteer

#### randy patton

##### Expertise

college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

##### Experience

26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

Publications
J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

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M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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Also an Expert in Oceanography