A plane bisects the line segment joining the | vedclass

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Question

(A) Let the points be $A(1, 2, 3)$ and $B(-3, 4, 5)$.Since the plane bisects the line segment $AB$ at right angles,it passes through the midpoint $M$

Vedclass · Accepted Answer

$(-3, 2, 1)$

Vedclass · Answer

(A) Let the points be $A(1, 2, 3)$ and $B(-3, 4, 5)$.Since the plane bisects the line segment $AB$ at right angles,it passes through the midpoint $M$ of $AB$.$M = \left(\frac{1-3}{2}, \frac{2+4}{2}, \frac{3+5}{2}\right) = (-1, 3, 4)$.The normal vector $\vec{n}$ to the plane is the vector $\vec{AB} = (-3-1, 4-2, 5-3) = (-4, 2, 2)$.We can simplify the normal vector to $\vec{n}' = (-2, 1, 1)$.The equation of the plane is $-2(x - (-1)) + 1(y - 3) + 1(z - 4) = 0$.$-2x - 2 + y - 3 + z - 4 = 0 \Rightarrow -2x + y + z = 9$.Now,check the options:For $(-3, 2, 1)$: $-2(-3) + 2 + 1 = 6 + 2 + 1 = 9$. This satisfies the equation.Thus,the plane passes through the point $(-3, 2, 1)$.

$A$ plane bisects the line segment joining the points $(1, 2, 3)$ and $(-3, 4, 5)$ at right angles. Then this plane also passes through the point

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