What is the reflection of the point P(1, 3, 4 | vedclass

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(B) Let the reflection of the point $P(1, 3, 4)$ in the given plane be $Q(x, y, z)$.The line passing through $P$ and perpendicular to the plane has th

Vedclass · Accepted Answer

$(-3, 5, 2)$

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(B) Let the reflection of the point $P(1, 3, 4)$ in the given plane be $Q(x, y, z)$.The line passing through $P$ and perpendicular to the plane has the direction ratios $(2, -1, 1)$.The equation of this line is $\frac{x - 1}{2} = \frac{y - 3}{-1} = \frac{z - 4}{1} = r$.Any point on this line is given by $(2r + 1, -r + 3, r + 4)$.Let $M$ be the midpoint of $PQ$. The coordinates of $M$ are $\left( \frac{2r + 1 + 1}{2}, \frac{-r + 3 + 3}{2}, \frac{r + 4 + 4}{2} \right) = (r + 1, -0.5r + 3, 0.5r + 4)$.Since $M$ lies on the plane $2x - y + z + 3 = 0$,we substitute the coordinates:$2(r + 1) - (-0.5r + 3) + (0.5r + 4) + 3 = 0$.$2r + 2 + 0.5r - 3 + 0.5r + 4 + 3 = 0$.$3r + 6 = 0 \implies r = -2$.Substituting $r = -2$ into the coordinates of $Q(2r + 1, -r + 3, r + 4)$:$x = 2(-2) + 1 = -3$.$y = -(-2) + 3 = 5$.$z = -2 + 4 = 2$.Thus,the reflection point $Q$ is $(-3, 5, 2)$.

What is the reflection of the point $P(1, 3, 4)$ in the plane $2x - y + z + 3 = 0$?

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