Understanding the Equation of a Line in 3D

The question asks for the direction cosines of a line that is parallel to a given line PQ. The equation of line PQ is provided in its symmetric form:

The standard symmetric form of the equation of a line passing through a point \((x_1, y_1, z_1)\) and having direction ratios \( (a, b, c) \) is:

To find the direction ratios of the given line PQ, we need to rewrite its equation to match the standard symmetric form.

Extracting Direction Ratios from the Equation

Let's rewrite the given equation:

The first part, \( \frac{x+1}{2} \), can be written as \( \frac{x-(-1)}{2} \). Here, the denominator is 2.

The second part is \( \frac{2-y}{5} \). To get the term \( (y-y_1) \) in the numerator, we can rewrite \( 2-y \) as \( -(y-2) \). So, \( \frac{2-y}{5} = \frac{-(y-2)}{5} = \frac{y-2}{-5} \). Here, the denominator is -5.

The third part is \( \frac{z+6}{7} \). This can be written as \( \frac{z-(-6)}{7} \). Here, the denominator is 7.

So, the symmetric form of the equation of line PQ is:

Comparing this with the standard form \( \frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c} \), we can identify the direction ratios \( (a, b, c) \) of the line PQ as \( (2, -5, 7) \).

Direction Ratios vs. Direction Cosines

• Direction Ratios: Three numbers \( a, b, c \) that are proportional to the direction cosines of a line. They are the components of a vector parallel to the line.
• Direction Cosines: The cosines of the angles that a line makes with the positive x, y, and z axes. If the direction ratios are \( (a, b, c) \), the direction cosines \( (l, m, n) \) are given by \( l = \frac{a}{\sqrt{a^2+b^2+c^2}}, m = \frac{b}{\sqrt{a^2+b^2+c^2}}, n = \frac{c}{\sqrt{a^2+b^2+c^2}} \). The sum of the squares of direction cosines is always 1 (\( l^2 + m^2 + n^2 = 1 \)).

Calculating the Direction Cosines of PQ

The direction ratios of line PQ are \( (a, b, c) = (2, -5, 7) \).

First, we calculate the magnitude of the direction vector, which is \( \sqrt{a^2 + b^2 + c^2} \):

Now, we can calculate the direction cosines \( (l, m, n) \):

So, the direction cosines of line PQ are \( \left(\frac{2}{\sqrt{78}}, \frac{-5}{\sqrt{78}}, \frac{7}{\sqrt{78}}\right) \).

Direction Cosines of a Parallel Line

Two lines are parallel if and only if their direction ratios are proportional, which means their direction cosines are the same or differ only by a sign change for all components (if the direction vector is in the opposite direction). However, the standard convention for direction cosines usually refers to a specific direction, so for a line parallel to PQ, its direction cosines will be the same as PQ's direction cosines, \( \left(\frac{2}{\sqrt{78}}, \frac{-5}{\sqrt{78}}, \frac{7}{\sqrt{78}}\right) \).

Comparing with Options

Let's check the given options:

1. \( \left(\frac{2}{\sqrt{78}}, \frac{5}{\sqrt{78}}, \frac{7}{\sqrt{78}}\right) \) - Incorrect. The sign of the second component is positive.
2. \( \left(\frac{-5}{\sqrt{78}}, \frac{-2}{\sqrt{78}}, \frac{7}{\sqrt{78}}\right) \) - Incorrect. The components are mixed up and signs are incorrect.
3. \( \left(\frac{5}{\sqrt{78}}, \frac{-2}{\sqrt{78}}, \frac{7}{\sqrt{78}}\right) \) - Incorrect. The components are mixed up.
4. \( \left(\frac{2}{\sqrt{78}}, \frac{-5}{\sqrt{78}}, \frac{7}{\sqrt{78}}\right) \) - Correct. This matches our calculated direction cosines.

Thus, the direction cosines of a line parallel to PQ are \( \left(\frac{2}{\sqrt{78}}, \frac{-5}{\sqrt{78}}, \frac{7}{\sqrt{78}}\right) \).

Revision Table: Key Concepts in 3D Lines

Additional Information on Direction Cosines and Parallelism

Direction cosines are fundamental to describing the orientation of a line in three-dimensional space. They uniquely define the direction of the line (up to a sign). If a line has direction cosines \((l, m, n)\), any line parallel to it will also have direction cosines \((l, m, n)\). This is because parallel lines point in the same direction, even if they pass through different points in space.

When given the equation of a line in symmetric form \( \frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c} \), the denominators \( a, b, c \) are the direction ratios of the line. It is crucial to ensure that the numerators are exactly \( (x-x_1), (y-y_1), \) and \( (z-z_1) \) with a positive sign for the variables \( x, y, z \). If a term like \( (c-y) \) appears, it must be rewritten as \( -(y-c) \) and the negative sign absorbed into the denominator, changing its sign.

For example, in the given problem, \( \frac{2-y}{5} \) was rewritten as \( \frac{y-2}{-5} \) to correctly identify the direction ratio for the y-component as -5, not 5. Failing to do this is a common mistake that leads to incorrect direction ratios and consequently, incorrect direction cosines.