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P(t) = (1-t, 2-3t, 2t-1), Q(t) = (1-t^2, 2-3t^2, 2t^2 -1)

Source: Spanish Mathematical Olympiad 1973 P8

December 23, 2022
analytic geometryvectorlinear algebra

Problem Statement

In a three-dimensional Euclidean space, by u1→\overrightarrow{u_1} , u2→\overrightarrow{u_2} , u3→\overrightarrow{u_3} are denoted the three orthogonal unit vectors on the x,yx, y, and zz axes, respectively. a) Prove that the point P(t)=(1−t)u1→+(2−3t)u2→+(2t−1)u3→P(t) = (1-t)\overrightarrow{u_1} +(2-3t)\overrightarrow{u_2} +(2t-1)\overrightarrow{u_3} , where tt takes all real values, describes a straight line (which we will denote by LL). b) What describes the point Q(t)=(1−t2)u1→+(2−3t2)u2→+(2t2−1)u3→Q(t) = (1-t^2)\overrightarrow{u_1} +(2-3t^2)\overrightarrow{u_2} +(2t^2 -1)\overrightarrow{u_3} if tt takes all the real values? c) Find a vector parallel to LL. d) For what values of tt is the point P(t)P(t) on the plane 2x+3y+2z+1=02x+ 3y + 2z +1 = 0? e) Find the Cartesian equation of the plane parallel to the previous one and containing the point Q(3)Q(3). f) Find the Cartesian equation of the plane perpendicular to LL that contains the point Q(2)Q(2).
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