Question

Equation of the plane through the mid-point of the line segment joining the points P(4, 5, -10) and Q(-1, 2, 1) and perpendicular to PQ is:

• A. \( \vec{r} \cdot \left(\frac{3}{2}\hat{i} + \frac{7}{2}\hat{j} - \frac{9}{2}\hat{k}\right) = 45 \)
• B. \( \vec{r} \cdot (-\hat{i} + 2\hat{j} + \hat{k}) = \frac{135}{2} \)
• C. \( \vec{r} \cdot (5\hat{i} + 3\hat{j} - 11\hat{k}) + \frac{135}{2} = 0 \)
• D. \( \vec{r} \cdot (4\hat{i} + 5\hat{j} - 10\hat{k}) = 85 \)
• E. \( \vec{r} \cdot (5\hat{i} + 3\hat{j} - 11\hat{k}) = \frac{135}{2} \)

Hint:

The normal vector of a plane determines the coefficients of $x, y,$ and $z$. If the plane is perpendicular to $PQ$, the coefficients must be proportional to the differences in coordinates of $P$ and $Q$.

Answer 1

Answer: (E)