Distance Between Two Skew Lines?

📌 CONCEPT: The shortest distance between two skew lines in a 3D space can be determined using the formula involving the direction cosines of the lines and a point on one of the lines.

📐 RULE / FORMULA: The formula to find the shortest distance between two skew lines is given by d = |(a1b2 - a2b1) / √((a1^2 + b1^2 + c1^2) * (a2^2 + b2^2 + c2^2))|, where (a1, b1, c1) and (a2, b2, c2) are the direction cosines of the lines and (a1b2 - a2b1) is the numerator.

💡 WORKED EXAMPLE: To find the shortest distance between L1 and L2, we first determine the direction cosines of the lines. Let's say the direction cosines of L1 are (1/√3, 1/√3, 1) and of L2 are (-1/√3, -1/√3, 1). The point P (2, 3, 4) lies on L1. Substituting these values in the formula, we get d = |(1/√3 * -1/√3 - 1 * -1/√3) / √((1/√3)^2 + (1/√3)^2 + 1^2 * (-1/√3)^2 + (-1/√3)^2 + 1^2)| = 1/√3.

⚠️ COMMON MISTAKE: Students often forget to substitute the correct direction cosines of the lines and the given point in the formula, leading to incorrect calculations of the shortest distance.