Question

Find the equation of the line segment from \(\mathbf{X}_{0}\) to \(\mathbf{X}_{1}\). (a) \(\mathbf{X}_{0}=(1,-3,4,2), \quad \mathbf{X}_{1}=(2,0,-1,5)\) (b) \(\mathbf{X}_{0}=(3,1-2,1,4), \quad \mathbf{X}_{1}=(2,0,-1,4,-3)\) (c) \(\mathbf{X}_{0}=(1,2,-1), \quad \mathbf{X}_{1}=(0,-1,-1)\)

Short answer

Write the equation of the line segment in each of the following cases: a) In 4D space, between points (1,-3,4,2) and (2,0,-1,5) b) In 5D space, between points (3,1-2,1,4) and (2,0,-1,4,-3) c) In 3D space, between points (1,2,-1) and (0,-1,-1) Solution: a) The equation of the line segment in 4D space is: \(\mathbf{X}(t)=(1+t, -3+3t, 4-5t, 2+3t)\) for \(0 \leq t \leq 1\). b) The equation of the line segment in 5D space is: \(\mathbf{X}(t)=(3-t,1-2-t,1-2t,4-7t)\) for \(0 \leq t \leq 1\). c) The equation of the line segment in 3D space is: \(\mathbf{X}(t)=(1-t,2-3t,-1)\) for \(0 \leq t \leq 1\).

Step-by-step solution

Case (a) - 4D space

We have points \(\mathbf{X}_{0}=(1,-3,4,2)\), and \(\mathbf{X}_{1}=(2,0,-1,5)\). We'll create the equation of the line segment connecting these points using the parametric equation: \(\mathbf{X}(t) = \mathbf{X}_{0} + t(\mathbf{X}_{1}-\mathbf{X}_{0})\). Find the difference vector \(\mathbf{X}_{1}-\mathbf{X}_{0}\) \(\mathbf{X}_{1}-\mathbf{X}_{0} = (2-1,0-(-3),-1-4,5-2) = (1,3,-5,3)\) Now substitute back into the parametric equation: \(\mathbf{X}(t) = (1,-3,4,2) + t(1,3,-5,3)\) Thus, the equation of the line segment in 4D space is: \(\mathbf{X}(t) = (1+t, -3+3t, 4-5t, 2+3t)\) for \(0 \leq t \leq 1\).

Case (b) - 5D space

We have points \(\mathbf{X}_{0}=(3,1-2,1,4)\), and \(\mathbf{X}_{1}=(2,0,-1,4,-3)\). We'll create the equation of the line segment connecting these points using the parametric equation: \(\mathbf{X}(t) = \mathbf{X}_{0} + t(\mathbf{X}_{1}-\mathbf{X}_{0})\). Find the difference vector \(\mathbf{X}_{1}-\mathbf{X}_{0}\) \(\mathbf{X}_{1}-\mathbf{X}_{0} = (2-3,0-(1-2),-1-1,-3-4) = (-1,-1,-2,-7)\) Now substitute back into the parametric equation: \(\mathbf{X}(t) = (3,1-2,1,4) + t(-1,-1,-2,-7)\) Thus, the equation of the line segment in 5D space is: \(\mathbf{X}(t) = (3-t,1-2-t,1-2t,4-7t)\) for \(0 \leq t \leq 1\).

Case (c) - 3D space

We have points \(\mathbf{X}_{0}=(1,2,-1)\), and \(\mathbf{X}_{1}=(0,-1,-1)\). We'll create the equation of the line segment connecting these points using the parametric equation: \(\mathbf{X}(t) = \mathbf{X}_{0} + t(\mathbf{X}_{1}-\mathbf{X}_{0})\). Find the difference vector \(\mathbf{X}_{1}-\mathbf{X}_{0}\) \(\mathbf{X}_{1}-\mathbf{X}_{0} = (0-1,-1-2,-1-(-1)) = (-1,-3,0)\) Now substitute back into the parametric equation: \(\mathbf{X}(t) = (1,2,-1) + t(-1,-3,0)\) Thus, the equation of the line segment in 3D space is: \(\mathbf{X}(t) = (1-t,2-3t,-1)\) for \(0 \leq t \leq 1\).

Key concepts

Line Segments in Multidimensional Space

Line segments in multidimensional space extend the familiar concept of a line in two- or three-dimensional space to spaces with four or more dimensions. This is crucial in fields like physics and computer science where higher-dimensional representations are needed.
A line segment here is defined between two points, starting at point \( \mathbf{X}_0 \) and ending at point \( \mathbf{X}_1 \). In multidimensional space, each point is represented as a vector with several components, one for each dimension. For example:

• In 3D, the point might be \( (x, y, z) \)
• In 4D, \( (x, y, z, w) \)
• And in 5D, \( (x, y, z, w, v) \)

A parametric equation is used to describe the line segment. By adjusting a parameter \( t \) (where \( 0 \leq t \leq 1 \)), you can move from the starting point \( \mathbf{X}_0 \) to the end point \( \mathbf{X}_1 \).
As \( t \) progresses from 0 to 1, the equation traces the path of the line segment.

Difference Vectors

Understanding difference vectors is essential when dealing with line segments in multidimensional space. A difference vector shows how you get from one point to another. It's like the travel direction and distance you need to cover in space.
To find a difference vector, you subtract each component of the start vector \( \mathbf{X}_0 \) from the corresponding component of the end vector \( \mathbf{X}_1 \). It works like this:

• Start with the end point \( \mathbf{X}_1 = (x_1, y_1, z_1) \)
• Subtract the starting point \( \mathbf{X}_0 = (x_0, y_0, z_0) \)
• Get a new vector: \( \mathbf{X}_1 - \mathbf{X}_0 = (x_1 - x_0, y_1 - y_0, z_1 - z_0) \)

This new vector points straight from \( \mathbf{X}_0 \) to \( \mathbf{X}_1 \). It's the backbone for creating the parametric equation of your line segment.
Remember, each subtraction operation gives you the component's direction and distance along each axis.

Vector Operations

Vector operations are the mathematical manipulations we perform on vectors, which are fundamental in expressing line segments, especially in multidimensional spaces. In the context of this exercise, we primarily focus on addition and scalar multiplication.
**Vector Addition:** This is used in forming the parametric equation for a line segment. By combining the starting point vector and a scaled-up difference vector, we create locations along the segment. For vectors \( \mathbf{A} = (a_1, a_2, a_3) \) and \( \mathbf{B} = (b_1, b_2, b_3) \), addition results in: (where each component is added to its counterpart):

• \( \mathbf{A} + \mathbf{B} = (a_1 + b_1, a_2 + b_2, a_3 + b_3) \)

**Scalar Multiplication:** It involves adjusting the length of a vector. By multiplying a vector \( \mathbf{A} \) by a scalar \( t \), you stretch or shrink it. Done on every component, it gives:

• \( t \mathbf{A} = (t a_1, t a_2, t a_3) \)

Both operations are key in forming the parametric equation: combining scalar-scaled difference vectors with the starting point vector to define positions along the line segment.
Mastering these operations lets you handle any changes in multidimensional space effectively.