When dealing with lines in mathematics, it is important to know how to find the vector equation and parametric equations for the line segment that joins two points. This is a fundamental concept in linear algebra and is used in a variety of applications, from computer graphics to physics.

Understanding the Line Segment

A line segment is a part of a line that is bounded by two distinct end points. In other words, it is a straight line that starts at one point and ends at another. To find the vector equation and parametric equations for the line segment that joins two points, we need to first understand what a vector equation and a parametric equation are.

Vector Equation

A vector equation is a way to express a line in terms of a direction vector and a point on the line. The direction vector is a vector that points in the direction of the line, while the point on the line is any point that lies on the line. The vector equation of a line is given by:

r = a + tb

where:

• r is a position vector that describes any point on the line
• a is a position vector that describes a point on the line
• b is a direction vector that describes the direction of the line
• t is a scalar value that can take any real number

Using this equation, we can find the vector equation for the line segment that joins two points.

Parametric Equations

Parametric equations are another way to express a line in terms of its coordinates. They are given by:

x = x₀ + at

y = y₀ + bt

z = z₀ + ct

where:

• x, y, and z are the coordinates of any point on the line
• x₀, y₀, and z₀ are the coordinates of a point on the line
• a, b, and c are the direction ratios of the line
• t is a scalar value that can take any real number

Parametric equations can also be used to find the vector equation for the line segment that joins two points.

Finding the Vector Equation and Parametric Equations for the Line Segment

Now that we understand what a vector equation and parametric equations are, we can use them to find the vector equation and parametric equations for the line segment that joins two points.

Let's say we have two points, p and q, and we want to find the vector equation and parametric equations for the line segment that joins them. The first step is to find the direction vector of the line segment. We can do this by subtracting the coordinates of one point from the coordinates of the other point.

b = q - p

Once we have the direction vector, we can choose any point on the line segment as our starting point. Let's choose p. We can then use the vector equation to find the vector equation for the line segment that joins p to q.

r = p + tb

To find the parametric equations, we can substitute the coordinates of the vectors into the parametric equation formula.

x = x_p + (x_q - x_p)t

y = y_p + (y_q - y_p)t

z = z_p + (z_q - z_p)t

where:

• x_p, y_p, and z_p are the coordinates of point p
• x_q, y_q, and z_q are the coordinates of point q
• t is a scalar value that can take any real number

Example

Let's say we have two points, p(1, 2, 3) and q(4, 5, 6), and we want to find the vector equation and parametric equations for the line segment that joins them.

The direction vector of the line segment is:

b = q - p = (4 - 1)i + (5 - 2)j + (6 - 3)k = 3i + 3j + 3k

Choosing p as our starting point, the vector equation for the line segment is:

r = p + tb = (1 + 3t)i + (2 + 3t)j + (3 + 3t)k

The parametric equations for the line segment are:

x = 1 + 3t

y = 2 + 3t

z = 3 + 3t

Conclusion

Understanding how to find the vector equation and parametric equations for the line segment that joins two points is an important skill in mathematics. It allows us to express lines in terms of their coordinates, which is useful in a variety of applications. By following the steps outlined in this article, you can easily find the vector equation and parametric equations for any line segment.

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