(a) Describe the line whose symmetric equations are (x-1)/(2)=(y+3)/(4)=z-5 (see Exercise 52 ).

step 1 The question we have been given a symmetric equation of line x minus 1 upon 2 is equal to 5 plus

(a) Describe the line whose symmetric equations are...

Key Concepts

Direction Vector

The direction vector of a line is a key component that indicates the line's orientation in space. In the symmetric equation, the components of the vector are the denominators (or the coefficients of t after conversion to parametric form), and they play a crucial role in determining parallelism, orthogonality, and the general geometry of the line.

Conversion Between Equation Forms

Converting between symmetric and parametric forms is a fundamental technique in analytic geometry. The process involves introducing a parameter t that equates each ratio from the symmetric form, thereby rewriting each coordinate as an expression in t. This conversion facilitates easier computation and provides a clearer view of the line's behavior in space.

Symmetric Equations

Symmetric equations of a line provide a compact representation by expressing the ratios of differences in coordinates relative to a point on the line, typically in the form (x - x?)/a = (y - y?)/b = (z - z?)/c. This form directly illustrates both a point through which the line passes and the direction vector, making it essential for describing lines in three-dimensional space.

Parametric Equations

Parametric equations describe a line by expressing each coordinate as a function of a single parameter, usually noted as t. This form, typically written as x = x? + at, y = y? + bt, and z = z? + ct, makes it easy to generate points on the line and is useful in various applications, including motion and intersection phenomena.

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