Let the line be given by x= ut, y= vt, z= wt (since (0,0,0) satifies that). If the plane and line are orthogonal, then <u, v, w> must be orthogonal to the plane and so must be parallel to <a, b, c>. That is u= ka, v=kb, w= kc. x= kat, y= kbv, z= kct. Putting that into the equation of the plane, [itex]k(a^2+ b^2+ c^2)t+ d= 0[/itex]. [itex]t= -d/(a^2+ b^2+ c^2)[/itex]. Put that into the equation of the line and solve for x, y, and z.Nikitin said:Homework Statement
A plane has the equation aX + bY + cZ + d = 0. A line L goes from (0,0,0) and crosses the plane at some point. L and plane are orthogonal. express the coordinates of the crossing point P by: a, b, c and d.
Homework Equations
I know that [a,b,c] is the directional vector for the line and that
|d|/sqrt(a^2 + b^2 + c^2) is the distance between point P and origo. I know and understand this at least.
The Attempt at a Solution
I'm lost lol
Nikitin said:ah damn, I haven't learned about directional vectors this year. but i was reminded about that stuff by a friend who helped me with this assignment.
So, if I divide [a,b,c] by the length of the directional vector, sqrt(a^2 + b^2 + c^2) and then multiply it by the length of the OP vector, |d|/sqrt(a^2 + b^2 + c^2), I get:
a|d|/a^2 + b^2 + c^2, b|d|/a^2 + b^2 + c^2, c|d|/a^2 + b^2 + c^2 = P.
The answer is supposed to be (-ad/a^2 + b^2 + c^2,-bd/a^2 + b^2 + c^2,-cd/a^2 + b^2 + c^2) = P
*sigh* does this have something to do with all the fools sometimes saying a plane's equation is ax+by+cz -d = 0 instead of ax+by+cz +d = 0 ? or that d is negative if the directional vector is positive?
has been a really long day at school + after 4 me and I'm sleepy as hell.. hope somebody can fill me in tomorrow
An extremely tough vectors-planes problem is a mathematical problem that involves finding the intersection point between two or more planes in three-dimensional space using vector equations. It often requires advanced mathematical skills and techniques to solve.
Vectors-planes problems can be difficult because they require a strong understanding of vector operations, as well as knowledge of plane geometry and algebraic techniques. They also often involve multiple steps and require a lot of calculation and manipulation.
First, make sure you have a solid understanding of vector operations and plane geometry. Then, carefully analyze the given information and determine which equations and techniques can be used to solve the problem. Work through the problem step by step, and double check your calculations and solutions.
Yes, some common mistakes include not fully understanding the given information, using incorrect equations or operations, and making calculation errors. It is important to carefully check your work and make sure all steps are correct.
One helpful tip is to draw a diagram or visualize the problem in three-dimensional space. This can make it easier to understand the given information and determine the correct approach for solving the problem. It is also important to practice and become familiar with different types of vectors-planes problems to improve problem-solving skills.