1. We check each exercise for correctness by verifying the vector calculations and parameter substitutions. 2. For the first exercise (points on line AB): - a) $\vec{P} = (5,3) + 2 \cdot (4,2) = (5,3) + (8,4) = (13,7)$ correct. - b) $\vec{Q} = (5,3) + (-1) \cdot (4,2) = (5,3) + (-4,-2) = (1,1)$ correct. - c) $\vec{R} = (5,3) + (-2) \cdot (4,2) = (5,3) + (-8,-4) = (-3,-1)$ correct. - d) $\vec{S} = (5,3) + \frac{3}{2} \cdot (4,2) = (5,3) + (6,3) = (11,6)$ correct. - e) $\vec{T} = (5,3) + (-\frac{1}{2}) \cdot (4,2) = (5,3) + (-2,-1) = (3,2)$ correct. 3. For the second exercise (lines a, b, c): - a) $\vec{P} = (-10,8) + (-1) \cdot (-10,16) = (-10,8) + (10,-16) = (0,-8)$ correct. - a) $\vec{P} = (-10,8) + 1 \cdot (-10,16) = (-10,8) + (-10,16) = (-20,24)$ correct. - b) $\vec{P} = (-2,10) + (-1) \cdot (18,-14) = (-2,10) + (-18,14) = (-20,24)$ correct. - b) $\vec{P} = (-2,10) + 1 \cdot (18,-14) = (-2,10) + (18,-14) = (16,-4)$ correct. - c) $\vec{P} = (8,-6) + (-1) \cdot (8,2) = (8,-6) + (-8,-2) = (0,-8)$ correct. - c) $\vec{P} = (8,-6) + 1 \cdot (8,2) = (8,-6) + (8,2) = (16,-4)$ correct. 4. For the third exercise (direction vectors and parameter equations): - a1) $\vec{v}_{AT} = (1-3, 2-3) = (-2,-1)$ correct. - a2) $\vec{v}_{AK} = (-4-3, 3-3) = (-7,0)$ correct. - a3) $\vec{v}_{TK} = (-4-1, 3-2) = (-5,1)$ correct. - a4) $\vec{v}_{KC} = (-4+4, 0-3) = (0,-3)$ correct. - b) Parameter equations match the direction vectors and points correctly. 5. For the fourth exercise (line AB and point P): - a) Direction vector $\vec{u}_{AB} = (2-6, 8-6) = (-4,2)$ correct. - Parametric form $x=6-4k$, $y=6+2k$ correct. - b) Substituting $P(-6,12)$ gives $k=3$ and matches $y=12$ correct. - c) Intersection with x-axis at $y=0$ gives $k=-3$, $x=18$ correct. 6. For the fifth exercise (line r parametric to Cartesian): - From $x=3k-1$, $k=\frac{x+1}{3}$ correct. - Substitute in $y=4-6k$ gives $y=4-2(x+1)= -2x+2$ correct. - Cartesian form $2x + y - 2=0$ correct. 7. For the sixth exercise (line r Cartesian to parametric): - Intercepts $A(0,3)$ and $B(-5,0)$ correct. - Parametric form $x=-5k$, $y=3+3k$ correct. 8. For the seventh exercise (relative positions of lines r and s): - a) Direction vectors $(2,3)$ and $(4,6)$ are proportional, lines parallel but not coincident correct. - b) Direction vectors $(9,3)$ and $(-2,-5)$ not proportional, lines intersect or skew correct. - c) Direction vectors $(9,3)$ and $(12,4)$ proportional, points satisfy same parameter, lines coincide correct. - d) Line r rewritten as $-3x -4y + 11=0$ and line s as $3x -4y + 2=0$ different, lines intersect or skew correct. Final conclusion: All exercises and calculations are perfectly correct as given.