Question

Equation of line passing through \((0,0,0)\) and parallel to Y-axis is (A) \((\mathrm{x} / 0) \overline{=(\mathrm{y}} / 1)=(\mathrm{z} / 1)\) (B) \((\mathrm{x} / 0)=(\mathrm{y} / 1)=(\mathrm{z} / 0)\) (C) \((\mathrm{x} / 0)=(\mathrm{y} / 0)=(\mathrm{z} / 1)\) (D) \((\mathrm{x} / 1)=(\mathrm{y} / 0)=(\mathrm{z} / 1)\)

Short answer

The short answer for the equation of the line passing through $(0,0,0)$ and parallel to the Y-axis is: \(\frac{x}{0}=\frac{y}{1}=\frac{z}{0}\).

Step-by-step solution

Understand the parametric form of a line in 3D space

The parametric equation of a line in 3D space is given by: \(\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}\) where (x0, y0, z0) is a point on the line, and (a, b, c) are the direction ratios of the line. In our case, the given point is the origin (0,0,0) and the line is parallel to the Y-axis.

Find the direction ratios for a line parallel to the Y-axis

For a line parallel to the Y-axis, the direction ratios should be in the form of (0, k, 0), where k is a non-zero constant.

Substitute the values of the point and direction ratios into the parametric equation

Given point is the origin (0,0,0) and direction ratios are (0, k, 0). Substituting, we get: \(\frac{x - 0}{0} = \frac{y - 0}{k} = \frac{z - 0}{0}\) This simplifies to: \(\frac{x}{0} = \frac{y}{k} = \frac{z}{0}\)

Compare the options and choose the correct one

Now, let's compare our result with all the given options: (A) \(\frac{x}{0}=\frac{y}{1}=\frac{z}{1}\) - This is not the same as our result. (B) \(\frac{x}{0}=\frac{y}{1}=\frac{z}{0}\) - This matches our result if we let k=1. (C) \(\frac{x}{0}=\frac{y}{0}=\frac{z}{1}\) - This is not the same as our result. (D) \(\frac{x}{1}=\frac{y}{0}=\frac{z}{1}\) - This is not the same as our result. Thus, the correct answer is option (B): \(\frac{x}{0}=\frac{y}{1}=\frac{z}{0}\).

Key concepts

Parametric Equations

In three-dimensional geometry, lines can be uniquely described using parametric equations. These equations express the coordinates of points on a line in terms of a parameter, often denoted by "t". This parameter allows us to capture the idea that each point on the line corresponds to a certain value of "t".
To write the parametric equation of a line, we consider a known point through which the line passes, say \( (x_0, y_0, z_0) \), and a direction vector \( (a, b, c) \) that indicates the direction of the line.

The parametric form of a line in 3D is given by:

• \( x = x_0 + at \)
• \( y = y_0 + bt \)
• \( z = z_0 + ct \)

These equations show how the position on the line relates to the parameter "t" and the direction vector. Thus, by varying "t", we traverse all points on the line.

Direction Ratios

Direction ratios are the components that define the direction of a line in space. For any vector or line defined in three-dimensional space, knowing the direction ratios helps determine its orientation.
Consider a line in 3D space described by its direction vector \( (a, b, c) \). These components serve as the direction ratios.

Key Features:

• The direction ratios are not unique; they can be any proportional values that give the same direction vector when scaled.
• For a line parallel to one of the coordinate axes, the direction ratios are particularly simple.
• For example, a line parallel to the Y-axis can have direction ratios like \( (0, k, 0) \), emphasizing that's it's only moving along the Y-axis.

This highlights that direction ratios are vital for expressing lines parallel to coordinate axes and they facilitate easier understanding of spatial orientation.

Lines in Three-Dimensional Space

Lines in three-dimensional (3D) space are pathways connecting an infinite set of points extending infinitely in both directions. Understanding lines in this space involves both parametric equations and direction ratios.
In 3D geometry, a line can be defined by two things:

• A fixed point through which it passes, such as \( (x_0, y_0, z_0) \).
• A direction vector \( (a, b, c) \) indicating its direction.

Lines can also be understood from their relationship with planes and axes:

• Lines parallel to axes: Simplifies to having zero direction ratios for some coordinates.
• Intersecting lines: Two lines intersect if they share a common point.
• Skew lines: Lines that do not intersect and are not parallel.

In essence, lines in 3D space are not just straight, they have orientation and position, making them an exciting concept at the intersection of algebra and geometry.