Line Equation: Point, Angle, And Values Of P & Q

Finding the equation of a line in three-dimensional space that passes through a given point and satisfies certain conditions is a fundamental problem in linear algebra and analytic geometry. In this article, we'll explore how to determine the equation of a line passing through the origin A(0, 0, 0) and another point P(p, 1, q). Furthermore, we'll delve into how to find the specific values of 'p' and 'q' when we know that the line forms a 45-degree angle with the x-axis. This involves understanding vector representation, direction vectors, and trigonometric relationships in 3D space.

Understanding the Basics

Before diving into the specifics, let's ensure we have a firm grasp of the underlying principles. The equation of a line in 3D space can be expressed in several forms, but the vector form is particularly useful for this problem. A line passing through a point A and parallel to a vector v can be described as:

r = a + tv

where:

• r is the position vector of any point on the line.
• a is the position vector of a known point on the line (in our case, A(0, 0, 0)).
• v is the direction vector of the line.
• t is a scalar parameter.

Determining the Direction Vector

In our scenario, the line passes through A(0, 0, 0) and P(p, 1, q). Therefore, the direction vector v can be found by subtracting the position vector of A from the position vector of P:

v = p - a = <p, 1, q> - <0, 0, 0> = <p, 1, q>

So, the direction vector v is simply <p, 1, q>. This vector dictates the orientation of the line in 3D space.

Equation of the Line

Now that we have the direction vector, we can write the equation of the line passing through A(0, 0, 0) and P(p, 1, q). Using the vector form, we have:

r = <0, 0, 0> + t<p, 1, q>

Which simplifies to:

r = t<p, 1, q> = <tp, t, tq>

This vector equation represents all points on the line. In component form, we can write this as:

x = tp y = t z = tq

These parametric equations describe the line in terms of the parameter 't'.

Finding p and q with the Angle Condition

The problem states that the line forms a 45-degree angle with the x-axis. This condition provides us with the information needed to determine the values of 'p' and 'q'. To understand how this works, we need to consider the direction cosines.

Direction Cosines

The direction cosines are the cosines of the angles that a vector makes with the coordinate axes. If α, β, and γ are the angles that the vector v = <p, 1, q> makes with the x, y, and z axes, respectively, then the direction cosines are:

cos(α) = p / ||v|| cos(β) = 1 / ||v|| cos(γ) = q / ||v||

where ||v|| is the magnitude of the vector v, given by:

||v|| = √(p² + 1² + q²) = √(p² + 1 + q²)

Applying the 45-Degree Angle Condition

We are given that the line forms a 45-degree angle with the x-axis, which means α = 45°. Therefore:

cos(45°) = p / √(p² + 1 + q²)

Since cos(45°) = √2 / 2, we have:

√2 / 2 = p / √(p² + 1 + q²)

Squaring both sides, we get:

1/2 = p² / (p² + 1 + q²)

This simplifies to:

p² + 1 + q² = 2p²

Which further simplifies to:

p² - q² = 1

Infinite Solutions and Additional Constraints

At this point, we have one equation with two unknowns, p and q:

p² - q² = 1

This equation represents a hyperbola in the p-q plane. It means that there are infinitely many pairs of (p, q) that satisfy this condition. To find specific values for p and q, we would need an additional constraint or piece of information.

Example of Additional Constraint

For instance, suppose we also knew that the line is perpendicular to the y-axis. This would mean that the angle β (the angle with the y-axis) is 90°. Thus:

cos(90°) = 1 / √(p² + 1 + q²)

0 = 1 / √(p² + 1 + q²)

This is impossible, indicating that the line cannot be simultaneously at a 45-degree angle to the x-axis and perpendicular to the y-axis, given that it passes through (0,0,0) and (p,1,q). Thus, there must be some other constraint. Suppose we were told q = 0, then

p² - 0² = 1 p² = 1 p = ±1

Thus, two such lines exist where P = (1, 1, 0) and P = (-1, 1, 0).

Conclusion

In summary, the equation of the line passing through A(0, 0, 0) and P(p, 1, q) is given by r = t<p, 1, q>. The values of p and q can be determined if we know that the line forms a 45-degree angle with the x-axis, leading to the equation p² - q² = 1. However, this single equation provides infinitely many solutions for p and q. To find specific values, additional constraints or information about the line's orientation in 3D space are required. This exploration demonstrates the interplay between vector algebra, analytic geometry, and trigonometric relationships in solving problems involving lines in three-dimensional space. Understanding these concepts is crucial for various applications in physics, engineering, and computer graphics.

Guys, remember that mastering these fundamental concepts opens doors to solving more complex problems in various fields. Keep practicing and exploring different scenarios to deepen your understanding!

Additional Considerations

Alternative Forms of the Line Equation

While we primarily focused on the vector form of the line equation, it's worth noting that other forms exist, such as the symmetric form and the parametric form. The symmetric form can be derived from the parametric equations by solving for 't' in each equation:

t = x/p = y/1 = z/q

This gives us the symmetric equation:

x/p = y = z/q

However, this form is only valid if p and q are non-zero. If either p or q is zero, the symmetric form is not applicable.

Impact of Different Angle Constraints

The 45-degree angle constraint with the x-axis significantly influences the possible values of p and q. If we had a different angle, say 60 degrees, the equation would change accordingly:

cos(60°) = p / √(p² + 1 + q²)

1/2 = p / √(p² + 1 + q²)

Squaring both sides:

1/4 = p² / (p² + 1 + q²)

p² + 1 + q² = 4p²

3p² - q² = 1

This new equation would lead to a different set of possible values for p and q, illustrating how sensitive the solution is to the given constraints.

Practical Applications

The problem of finding the equation of a line and determining its parameters based on angle constraints has numerous practical applications. In computer graphics, it's essential for rendering 3D objects and calculating lighting effects. In robotics, it's used for path planning and navigation. In physics, it's applied to describe the motion of particles in space.

For example, consider a robot arm moving in 3D space. If we know the starting point of the arm (A) and the desired end point (P), we can determine the equation of the line representing the arm's trajectory. By imposing angle constraints, we can ensure that the arm avoids obstacles or follows a specific path. Similarly, in computer graphics, when rendering a scene, we need to calculate the intersection of light rays with objects. This involves finding the equation of the light ray and determining its parameters based on its direction and origin.

Further Exploration

To deepen your understanding of this topic, consider exploring the following:

1. Direction Vectors and Direction Cosines: Study the properties of direction vectors and direction cosines in more detail. Understand how they relate to the angles that a line makes with the coordinate axes.
2. Vector Algebra: Review the fundamentals of vector algebra, including vector addition, subtraction, and scalar multiplication. These operations are essential for manipulating vectors and finding the equation of a line.
3. Analytic Geometry: Explore the connection between algebra and geometry in three-dimensional space. Understand how geometric objects, such as lines and planes, can be represented using algebraic equations.
4. Applications: Investigate real-world applications of line equations and angle constraints in various fields, such as computer graphics, robotics, and physics.

By delving deeper into these areas, you'll gain a more comprehensive understanding of the concepts discussed in this article and be better equipped to tackle related problems.

Remember, the key to mastering these concepts is practice. Work through various examples and exercises to solidify your understanding. Don't hesitate to seek help from textbooks, online resources, or instructors if you encounter difficulties. With perseverance and dedication, you'll become proficient in solving problems involving lines in three-dimensional space.

So keep up the great work, and remember, math is your friend!