Understanding the Condition for Perpendicular Lines

The question asks for the condition under which two lines, given by their parametric equations, are perpendicular. To determine if two lines are perpendicular, we need to look at their direction vectors. If the lines are perpendicular, their direction vectors must also be perpendicular. The condition for two vectors to be perpendicular is that their dot product is zero.

Finding Direction Vectors from Parametric Equations

The given lines are in the form:

Line 1: \(x = ay + b, z = cy + d\)

Line 2: \(x = a'y + b', z = c'y + d'\)

These equations represent lines in 3D space where 'y' acts as the parameter. We can rewrite these into a more standard parametric form \(\vec{r} = \vec{r}_0 + t\vec{v}\), where \(\vec{v}\) is the direction vector.

For Line 1:

• Let \(y = t\).
• Then \(x = at + b\) and \(z = ct + d\).
• A point on the line can be written as \((x, y, z) = (at + b, t, ct + d)\).
• We can express this as \((b, 0, d) + t(a, 1, c)\).
• Here, \((b, 0, d)\) is a point on the line, and \((a, 1, c)\) is the direction vector. Let's call the direction vector of Line 1 \(\vec{v}_1 = (a, 1, c)\). These equations are essentially parametric equations with parameter y.

For Line 2:

• Similarly, let \(y = t'\) for the second line.
• Then \(x = a't' + b'\) and \(z = c't' + d'\).
• A point on the line can be written as \((x, y, z) = (a't' + b', t', c't' + d')\).
• We can express this as \((b', 0, d') + t'(a', 1, c')\).
• Here, \((b', 0, d')\) is a point on the line, and \((a', 1, c')\) is the direction vector. Let's call the direction vector of Line 2 \(\vec{v}_2 = (a', 1, c')\).

Applying the Dot Product Condition for Perpendicularity

Two lines are perpendicular if and only if their direction vectors are perpendicular. The condition for two vectors to be perpendicular is that their dot product is zero.

The dot product of \(\vec{v}_1\) and \(\vec{v}_2\) is given by:

\(\vec{v}_1 \cdot \vec{v}_2 = (a, 1, c) \cdot (a', 1, c')\)

Calculating the dot product:

\(\vec{v}_1 \cdot \vec{v}_2 = a \cdot a' + 1 \cdot 1 + c \cdot c'\)

\(\vec{v}_1 \cdot \vec{v}_2 = aa' + 1 + cc'\)

For the lines to be perpendicular, this dot product must be zero:

\(aa' + 1 + cc' = 0\)

Rearranging the terms, the condition for these Perpendicular Lines is:

\(aa' + cc' + 1 = 0\)

This is the required condition for the given pair of lines to be perpendicular. This condition arises directly from the property that the dot product of the direction vectors must be zero for Perpendicular Lines.

Let's compare this condition with the given options:

• Option 1: \(aa' + cc' + 1 = 0\)
• Option 2: \(aa' + cc' - 1 = 0\)
• Option 3: \(ac + a'c' -1 = 0\)
• Option 4: \(ac + a'c' + 1 = 0\)

Our derived condition matches Option 1. Thus, for the given lines to be Perpendicular Lines, the relationship between the coefficients must be \(aa' + cc' + 1 = 0\).