Determining the Standard Equation of a Line Through a Given Point with Direction Ratios

Suppose we are interested in finding the standard equation of a line that passes through a specific point and has particular direction ratios. This problem is a fundamental concept in vector and analytic geometry. Specifically, we are given a line passing through the point (-1, 2, 3) with direction ratios 1, 0, -2. This article will detail how to derive the standard equation of this line, offering a clear understanding of the process involved.

Understanding Vector and Cartesian Equations of a Line

The vector equation of a line passing through a point with position vector (vec{a}) and parallel to a direction vector (vec{b}) can be expressed as:

[vec{r} vec{a} lambda vec{b}]

Here, (lambda) is a scalar parameter that varies along the line. This equation effectively describes the position of any point on the line through the point of origin plus a scaled version of the direction vector.

The Cartesian equation of a line passing through a point ((x_1, y_1, z_1)) with direction ratios (a, b, c) is given by:

[frac{x - x_1}{a} frac{y - y_1}{b} frac{z - z_1}{c}]

This form of the equation is useful in determining the coordinates of points on the line. Note that if (b 0), the equation simplifies to:

[frac{x - x_1}{a} frac{z - z_1}{c} y y_1]

Deriving the Standard Equation for the Given Line

Given the point (-1, 2, 3) and direction ratios 1, 0, -2, we can use the Cartesian form of the line equation. The standard equation of a line passing through a point ((-1, 2, 3)) and with direction ratios 1, 0, -2 is:

[frac{x 1}{1} frac{y - 2}{0} frac{z - 3}{-2}]

Since the direction ratio in the y-direction is 0, this indicates that the line is parallel to the xz-plane and does not change in the y-coordinate. Therefore, the y-coordinate of any point on the line remains constant at 2.

Summary and Further Exploration

In conclusion, we have derived the standard equation of a line that passes through the point (-1, 2, 3) and has direction ratios 1, 0, -2. This process combines both vector and Cartesian approaches to geometry, highlighting the flexibility and versatility of these mathematical tools.

For further study, one can explore how different points and direction ratios affect the line's equation, and how these equations can be applied in more complex geometric problems. The understanding of these basic principles is crucial for advanced topics in vector analysis and analytic geometry.