Vectors Problem Help
In a coordinate system in space are given points P (2, 3, 2), Q (3, 4, 1), R (5, 6, -1) and S (2, 4, 0).
a) Explain that points P, Q and R are on a straight line
b) Check if point S is also on that line.
c) Check if the line l containing points P and Q is parallel to the plane α: 3x + y + 4z + 1 = 0.
d) Determine the coordinates to the point where line l intersects xy plane.
The vector from the point $(a,b,c)$ to the point $(a',b',c')$ is $[a'-a,b'-b,c'-c].$ Two vectors $[u_1,u_2,u_3]$ and $[v_1,v_2,3_3]$ are proportional if $$[u_1,u_2,u_3]=c[v_1,v_2,3_3]$$ for some $c \ne 0.$ Three distinct points $P,Q,R$ are collinear if the vectors $PQ$ and $PR$ are proprtional. If the vector from the point $P$ to the point $Q$ is $[u_1,u_2,u_3]$, the condition for the line joining $P$ and $Q$ to be parallel to the plane $Ax+By+Cz+D=0$ is $$[A,B,C]\bullet[u_1,u_2,u_3]=0.$$ Parametric equations of the line $\ell$ containing the points $(a,b,c)$ and $(a',b',c')$ are $$x=a+(a'-a)t,y=b+(b'-b)t,z=c+(c'-c)t.$$ To find the point of intersection of the line $\ell$ with the plane $Ax+By+Cz+D=0$ first put the parametric expressions for $x,y$ and $z$ into the equation for the plane and solve for $t.$ Then put this value of $t$ into the parametric expressions for $x,y$ and $z$, which will give the coordinates of the point of intersection.