Question

Equation of line passes through \((-3,4,7)\) with direction \((5,2,8)\) is (A) \([(x-3) \overline{/ 5}]=[(y-4) / 2]=[(z-7) / 8]\) (B) \([(x+3) / 5]=[(y-4) / 2]=[(z-7) / 8]\) (C) \(x-3=y-4=z-7\) (D) \(x+3=y-4=z-7\)

Short answer

The short answer is: \( (B) \, \frac{x + 3}{5} = \frac{y - 4}{2} = \frac{z - 7}{8} \)

Step-by-step solution

Identify the given point and direction vector

We are given a point P(-3, 4, 7) and a direction vector D(5, 2, 8).

Substitute the coordinates and direction vector components into the equation

Now, substitute the coordinates of point P and the components of direction vector D into the point-direction form of the line equation: \[ \frac{x - (-3)}{5} = \frac{y - 4}{2} = \frac{z - 7}{8} \]

Simplify the equation

Next, simplify the equation by removing the negative sign from the x-coordinate: \[ \frac{x + 3}{5} = \frac{y - 4}{2} = \frac{z - 7}{8} \]

Match the equation with the given options

Now compare our simplified equation with the given options. Our equation matches option (B), which is: \[ \frac{x + 3}{5} = \frac{y - 4}{2} = \frac{z - 7}{8} \] Thus, the correct answer is (B).

Key concepts

Direction Vectors

Direction vectors play a vital role in defining lines in three-dimensional space. In coordinate geometry, a direction vector provides a way to describe the orientation of a line by specifying how the line moves away from a particular point in each dimension. For instance, if we have a direction vector \(D(a, b, c)\), it indicates that for each unit of length we move along the line, we move 'a' units in the x-direction, 'b' units in the y-direction, and 'c' units in the z-direction.

When solving problems involving the equation of a line, identifying the direction vector is a crucial step. In the provided exercise, the direction vector is \(D(5, 2, 8)\), meaning the line moves 5 units along the x-axis, 2 units along the y-axis, and 8 units along the z-axis for each interval. This vector is essential when formulating the equation of the line using the point-direction form.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry that uses a coordinate system to translate geometric figures and problems into algebraic equations. This approach allows for the resolution of geometric problems using algebraic methods.

In the context of three-dimensional space, coordinate geometry uses the three axes—x, y, and z—to form a three-dimensional coordinate system. Each point in space can be described by a set of coordinates \(P(x, y, z)\), representing its position relative to the origin \(O(0, 0, 0)\). To describe the equation of a line in three-dimensional space, we take a known point on the line and a direction vector that describes the line's path. This equation can be displayed in various forms, including the symmetric equations and the parametric form.

Point-Direction Form

The point-direction form is a way to express the equation of a line in space when you are given a point on the line and its direction vector. It emphasizes the relationship between a fixed point and a moving point along the line, dictated by the direction vector.

To write a line in point-direction form in three dimensions, we use the following structure \[ \frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c} \] where \(P(x_0, y_0, z_0)\) is a point on the line and the direction vector is \(D(a, b, c)\). Each ratio corresponds to how the coordinates change with respect to each other as you move along the line determined by the direction vector. The exercise presented shows how this form is applied by substituting the known point and direction vector into the form, leading to the correct answer option (B). Understanding this form is essential for describing lines in three dimensions and is a foundational concept in higher-dimensional geometry.