Question

The line \(x=a y+b, z=c y+d\) and \(x=a^{\prime} y+b^{\prime}, z=c^{\prime} y+d^{\prime}\) are perpendicular if (a) \(a a^{\circ}+c c^{\prime}=1\) (b) \(a a^{\circ}+c c^{\prime}=-1\) (c) \(b b^{2}+d d^{\prime}=1\) (d) \(b b^{\prime}+d d^{\prime}=-1\).

Short answer

The lines are perpendicular when \(a a' + c c' = 0\), but none of the provided options are correct.

Step-by-step solution

Understand the Problem

We want to determine the condition for two given lines to be perpendicular. We are given two lines in parametric form: \( x = ay + b, z = cy + d \) and \( x = a' y + b', z = c' y + d' \).

Recall the Perpendicular Condition for Lines

The condition for two lines (in vector form) to be perpendicular is that their direction vectors' dot product equals zero. If we assume the direction vectors for our lines as \((a, c)\) and \((a', c')\), then the condition becomes \((a, c) \cdot (a', c') = 0\).

Determine Direction Vectors

For the line \(x = ay + b, z = cy + d\), the direction vector is \((a, c)\). Similarly, for the line \(x = a'y + b', z = c'y + d'\), the direction vector is \((a', c')\).

Calculate Dot Product

Calculate the dot product of the direction vectors: \[ a \cdot a' + c \cdot c' \].

Set the Dot Product to Zero

According to the condition for perpendicularity, set the dot product equal to zero:\[ a \cdot a' + c \cdot c' = 0 \].

Match the Condition with Given Options

Compare the condition derived in Step 5 with the options provided in the exercise:- (a) \(a a^{ eflectbox{′}} + c c^{ eflectbox{′}} = 1\)- (b) \(a a^{ eflectbox{′}} + c c^{ eflectbox{′}} = -1\)- (c) \(b b^{ eflectbox{′}} + d d^{ eflectbox{′}} = 1\)- (d) \(b b^{ eflectbox{′}} + d d^{ eflectbox{′}} = -1\)Since we need \(a a' + c c' = 0\), none of the options are correct, and the provided options do not include zero as a choice.

Key concepts

Dot Product

The dot product is a fundamental operation in vector algebra that combines two vectors to produce a scalar. This operation requires both vectors to be in the same dimension and is often used to determine the angle between them or check perpendicularity.

To compute the dot product of two vectors, \(\vec{u} = (u_1, u_2, ..., u_n)\) and \(\vec{v} = (v_1, v_2, ..., v_n)\), you multiply the corresponding components and sum them up. This is given by the formula:

• \(\vec{u} \cdot \vec{v} = u_1 \cdot v_1 + u_2 \cdot v_2 + ... + u_n \cdot v_n\)

If the dot product of two vectors equals zero, the vectors are orthogonal, or perpendicular, which is crucial in solving problems involving perpendicular lines.

In the exercise, the given lines have direction vectors represented by the parameters related to \(y\). By calculating the dot product of these direction vectors and setting it to zero, we can determine if the lines are perpendicular.

Perpendicular Lines

Perpendicular lines intersect at a right angle (90 degrees). In vector terms, two lines are perpendicular if their direction vectors have a dot product of zero.

For example, consider two lines with vector directions \((a, c)\) and \((a', c')\). To determine if these lines are perpendicular, you calculate the dot product \(a \cdot a' + c \cdot c'\). If this equals zero, the lines are perpendicular.

• Perpendicular direction: \(\vec{u} \cdot \vec{v} = 0\)

In practical problems, ensuring this condition is met confirms the geometric relationship between the two lines. However, as seen in the problem, incorrectly assuming the result could lead to potential mismatches with the expected outcomes or options.

Direction Vectors

Direction vectors provide a way of describing the "path" or direction a line takes in space. In parametric equations, direction vectors help to express this path by capturing the coefficients of the variable (usually \(t\) or \(y\) in 2D/3D space).

For the line \(x = ay + b, z = cy + d\), the direction vector is \((a, c)\), indicating movement in a pattern associated with those coefficients.

• Represents direction: \((a, c)\) for the first line, \((a', c')\) for the second line.
• Makes the analysis of line interaction possible through dot products.

Understanding direction vectors is vital when assessing line relationships, such as intersection or parallelism, as they quantify how each line stretches through space.

Parametric Equations

Parametric equations allow a more flexible approach to defining lines, particularly in relation to their geometric properties. This type of equation separates a movement into component parts, allowing each part of the motion or line to be expressed as a function of one parameter (often \(y\) or \(t\)).

Consider these parametric equations: \(x = ay + b\) and \(z = cy + d\). Here, \(y\) acts as a parameter controlling how both \(x\) and \(z\) are determined, offering a formulaic way to set a line's path.

• Expresses lines in terms of a single parameter.
• Allows exploration of line properties such as direction and intersection.

Using parametric equations, one can manipulate the location and orientation of lines in more complex geometry problems, making them a potent tool in vector algebra.