Question

Find the equations of the lines AC and BD, if coordinates of points A, B, C and D respectively, are \((3, -4, 2)\), \((-1, -2, 0)\), \((4, 1, -1)\) and \((4, 0, 5)\). Show that the lines AC and BD are perpendicular to each other.

Hint:

Find \(\overrightarrow{AC} = (1, 5, -3)\) and \(\overrightarrow{BD} = (5, 2, 5)\). Their dot product is \(0\).

Answer 1

Step 1: Find the direction vectors:
Subtract the position vectors of the end points.
$\overrightarrow{AC} = \vec{C} - \vec{A} = (4\hat{i} + \hat{j} - \hat{k}) - (3\hat{i} - 4\hat{j} + 2\hat{k}) = \hat{i} + 5\hat{j} - 3\hat{k}$.
$\overrightarrow{BD} = \vec{D} - \vec{B} = (4\hat{i} + 5\hat{k}) - (-\hat{i} - 2\hat{j}) = 5\hat{i} + 2\hat{j} + 5\hat{k}$.

Step 2: Write line AC through C:
Any point of the line can be the base point. Take C this time.
\[ \vec{r} = 4\hat{i} + \hat{j} - \hat{k} + t(\hat{i} + 5\hat{j} - 3\hat{k}) \]
Cartesian form: \[ \frac{x-4}{1} = \frac{y-1}{5} = \frac{z+1}{-3} \]

Step 3: Write line BD through D:
For BD take the base point D.
\[ \vec{r} = 4\hat{i} + 5\hat{k} + s(5\hat{i} + 2\hat{j} + 5\hat{k}) \]
Cartesian form: \[ \frac{x-4}{5} = \frac{y-0}{2} = \frac{z-5}{5} \]
These describe the same lines as the ones through A and B, because the direction is the same and the points lie on the lines.

Step 4: Use the angle between the lines:
The angle $\theta$ between the lines satisfies $\cos\theta = \dfrac{\overrightarrow{AC} \cdot \overrightarrow{BD}}{|\overrightarrow{AC}||\overrightarrow{BD}|}$.
The dot product is $(1)(5) + (5)(2) + (-3)(5) = 5 + 10 - 15 = 0$.
The lengths are $|\overrightarrow{AC}| = \sqrt{35}$ and $|\overrightarrow{BD}| = \sqrt{54}$, and neither is zero.
So $\cos\theta = 0$ and $\theta = 90^{\circ}$. The lines AC and BD are perpendicular.

Final Answer:
Lines AC and BD are as written above, and since the dot product of their direction vectors is $0$, they are perpendicular.