Question 75
In a three-dimensional Euclidean space, the three orthogonal unit vectors on the axes $x, y, z$, respectively, are designated by $u_{1}, u_{2}, u_{3}$.
a) Proving that the point $P(t)=(1-t) u_{1}+(2-3 t) u_{2}+(2 t-1) u_{3}$, where $t$ takes all the real values, describes a line (which we will designate by $L$ ).
b) What describes the point $Q(t)=\left(1-t^{2}\right) u_{1}+\left(2-3 t^{2}\right) u_{2}+\left(2 t^{2}-1\right) u_{3}$ if $t$ takes all the actual values?
c) Find a parallel vector to $L$.
d) What values of $t$ is the $P(t)$ point on the $2 x+3 y+2 z+1=0$ plane?
e) Find the Cartesian equation of the plane parallel to the previous one and that contains the point $P(3)$. f) Find the Cartesian equation of the plane perpendicular to $L$ that contains the point $P(2)$.
a) Proving that the point $P(t)=(1-t) u_{1}+(2-3 t) u_{2}+(2 t-1) u_{3}$, where $t$ takes all the real values, describes a line (which we will designate by $L$ ).
b) What describes the point $Q(t)=\left(1-t^{2}\right) u_{1}+\left(2-3 t^{2}\right) u_{2}+\left(2 t^{2}-1\right) u_{3}$ if $t$ takes all the actual values?
c) Find a parallel vector to $L$.
d) What values of $t$ is the $P(t)$ point on the $2 x+3 y+2 z+1=0$ plane?
e) Find the Cartesian equation of the plane parallel to the previous one and that contains the point $P(3)$. f) Find the Cartesian equation of the plane perpendicular to $L$ that contains the point $P(2)$.