The plane which bisects the line segment join | vedclass

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(D) Let the points be $A(-3, -3, 4)$ and $B(3, 7, 6)$.The midpoint of the line segment $AB$ is $M = \left( \frac{-3+3}{2}, \frac{-3+7}{2}, \frac{4+6}{

Vedclass · Accepted Answer

$(4, 1, -2)$

Vedclass · Answer

(D) Let the points be $A(-3, -3, 4)$ and $B(3, 7, 6)$.The midpoint of the line segment $AB$ is $M = \left( \frac{-3+3}{2}, \frac{-3+7}{2}, \frac{4+6}{2} \right) = (0, 2, 5)$.The normal vector $\vec{n}$ to the plane is the vector $\overrightarrow{AB} = (3 - (-3))\hat{i} + (7 - (-3))\hat{j} + (6 - 4)\hat{k} = 6\hat{i} + 10\hat{j} + 2\hat{k}$.The equation of the plane is given by $\vec{r} \cdot \vec{n} = \vec{M} \cdot \vec{n}$.$\vec{r} \cdot (6\hat{i} + 10\hat{j} + 2\hat{k}) = (0\hat{i} + 2\hat{j} + 5\hat{k}) \cdot (6\hat{i} + 10\hat{j} + 2\hat{k})$.$6x + 10y + 2z = 0(6) + 2(10) + 5(2) = 20 + 10 = 30$.Dividing by $2$,we get the equation of the plane: $3x + 5y + z = 15$.Now,check the given options:For $(4, 1, -2)$: $3(4) + 5(1) + (-2) = 12 + 5 - 2 = 15$. This satisfies the equation.Therefore,the plane passes through the point $(4, 1, -2)$.

The plane which bisects the line segment joining the points $(-3, -3, 4)$ and $(3, 7, 6)$ at right angles,passes through which one of the following points?

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