Help with Vectors Please :(

[Lucid]

Hey guys, 

I'm studying for the May 2019 IB exams and I'm currently stuck on vectors. I was wondering how you find a perpendicular and/or parallel vector to a 3D vector? 

For example. 

"Find an equation of the line perpendicular to vector (3, 0, -1) and passing through point (4, 2, 1)." 

or, 

"Find the equation of the line parallel to vector (3, -2, 8), and passing through (3, 1, -2)" 

 

Thanks in advance  

[kw0573]

Parallel vectors have directions that are scalar multiples of each other. So the parallel line is r = s(3,-2,8) + (3,1,-2). It's pretty much analogous to y = mx  + b, in which you have the slope multiplied by a factor, and added to the y-intercept. 

As for perpendicular vectors, they have a dot product of 0. So for example (3, 0, -1) • (0, 1, 0) = 0, because 3(0) + 0(1) + (-1)(0) = 0. There are infinite perpendicular vectors to any vector. One possible perpendicular line is r = s(0, 1, 0) + (4, 2, 1)

These are the vector form of a line, which are accepted unless otherwise stated in the question.