Algebra - Vector Spaces - Analytic Geometry - Octave

Question

I'm studying for my algebra exam and I'm having problem solving these two simple True or False problems. It is a practice exam and I'm using Octave to solve them.

For the second exercise I think I need to use the command dot(r, s) in octave and to be 'true' the result must be 0 I guess, but I'm having problems with the 'r' equation.

$r=[ ? ]$

$s=[1, -1, 3]$ i guess? My though "$s:(x, y, z)= (2, 1, 2) + t(1, -1, 3), t∈R$"

dot(r, s) = ?

for the first question I have no idea what to do.

Thanks in advance!

Answer 1

I'm not very familiar with the syntax of Octave, but I suppose you'll be able to use Octave if you know what to do (mathematically).

For the first question:

• it should be clear that $(6,-3,9)$ is indeed a normal vector of the plane, since a plane with equation $ax+by+cz+d=0$ has $(a,b,c)$ as a normal vector;
• there are different ways of finding a direction vector of the line when it is given as the intersection of two planes: solve to get a parametric form of the line or take the cross product of the normal vectors of the two planes to see that $(0,1,1)$ is indeed a direction vector of $r$: $$(1,0,0) \times (0,1,-1) = (0,1,1)$$

For the second question: the lines are concurrent if they intersect in exactly one point. One way of checking this would be to substitute the parametric equations of line $s$ into the system of line $r$ and verify if this has a unique solution (for the parameter $t$).