Simple Cartesian but I can't bloomin work it out Watch
First year undergrad Engineering and we have been given a sheet of vector questions to do, most of them are stuff we have covered at A-level and the others just take a bit of inginuity, however they keep giving the equations for vectors in "full" cartesian form where the whole vector is just one equation (x+y+z=a )and I can't work out how to sort them into individual i,j,k (r=a+lambaB) equations that I need. I know how to get from the i,j,k equations to the one big equation however not back again.
For example a question was to find the shortest distance between a given point and a line x+2y-2z=9 , if I can convert the equation into individual equations for i,j and z then its just an A-level question that I can solve happily. I have tried googling online and looking through my textbook, however the textbook doesn't seem to even go into this level of complexity and when searching online I can't think of the right words to give me the results I want.
Any help would be awesome
To convert cartesian -> vector form, you need either two vectors or three points that lie on the plane.
The equation x+2y-2z=9 is the equation of a plane. Vectors associated with this plane are
a) a vector normal to the plane
b) vector(s) in the plane.
By inspection, a normal to the plane is <1, 2, -2> (the coefficients of x-y-z-).
So n = 1i +2j -2 k
If we find two linearly independent vectors in the plane, the we can form any vector in the plane as a linear combination of these vectors.
Surely you know how to find two linearly independent vectors in the plane. Just find three points on the plane simply by selecting arbitrary values for x and y and solving for z. Then continue. Or you could the null space method.
This question could be alternatively done using calculus.
^ This shows how to do a similar question
A vector equation answer may not be unique. Even if your vector equation is different to the textbook's answer etc it may not be actually wrong