Contradiction. Hence, no such vector $\mathbf{w}$ exists.
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- Substitute into (3):
- -2(3w_3 - 4) - 3(-1 - 2w_3) = 2 \
- -6w_3 + 8 + 3 + 6w_3 = 2 \Rightarrow 11 = 2
- oxed{ ext{No solution exists}}
- Question: A museum curator is restoring a triangular panel from a historical instrument, where the triangle has vertices at $A = (1, 3, -2)$, $B = (4, 1, 1)$, and $C = (2, -1, 3)$. Find the point $P$ on the line through $A$ and $B$ that is closest to point $C$.
- Solution: The line through $A$ and $B$ is parametrized as:
