Given the following vector equation in three dimensions r+(r xx d)=c where bb(c) and bb(d) are fixed given vectors, how can you find all solutions for bb(r)?

Bierlehre59

Bierlehre59

Answered question

2022-08-12

How to solve the vector equation
Given the following vector equation in three dimensions
r + ( r × d ) = c
where c and d are fixed given vectors, how can you find all solutions for r?
So far I have tried the following steps, to show that r must lie in a given plane.
By taking the dot product with d on both sides, we obtain
r + ( r × d ) = c ( r + ( r × d ) ) d = c d .
Since r × d is perpendicular to d, their dot product is zero, so we get the following (Equation ∗)
r d = c d
from which we can deduce that r (as a position vector) lies in the plane that contains c and is normal to the vector d.
However this doesn't necessarily imply that all points in this plane are valid solutions for r. I can't see how Equation (∗) can be substituted back into the original equation to somehow eliminate a term in r or simplify it. How do you solve this equation, making sure that you find all solutions for r?

Answer & Explanation

Malcolm Good

Malcolm Good

Beginner2022-08-13Added 14 answers

You have 3 direction in space. Assuming d and c are not collinear (in this case r would be in the same direction), then we can use the three directions to be c , d , and c × d . The first two might not be perpendicular. Then write
r = α c + β d + γ c × d
Plug this into your equation, then multiply the equation by each of the vectors of the basis.

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