Question

Suppose that \(\mathbf{U} \neq \mathbf{0}\) and \(\mathbf{V} \neq \mathbf{0}\). Complete the sentence: The equations $$ \mathbf{X}=\mathbf{X}_{0}+t \mathbf{U}, \quad-\infty

Short answer

Answer: The equations represent the same line in $\mathbb{R}^n$ if and only if their direction vectors are proportional (i.e., $\mathbf{U} = k\mathbf{V}$ for some scalar $k$), and there exists some values of $t$ and $s$ such that $\mathbf{X}_0 + t\mathbf{U} = \mathbf{X}_1 + s\mathbf{V}$.

Step-by-step solution

Analyze the two given equations

The given equations are $$ \mathbf{X}=\mathbf{X}_{0}+t \mathbf{U}, \quad-\infty<t<\infty $$ and $$ \mathbf{X}=\mathbf{X}_{1}+s \mathbf{V}, \quad-\infty<s<\infty $$ These equations represent lines in \(\mathbb{R}^{n}\) passing through points \(\mathbf{X}_0\) and \(\mathbf{X}_1\) with direction vectors \(\mathbf{U}\) and \(\mathbf{V}\), respectively.

Identify the condition that they represent the same line

In order for the equations to represent the same line, three conditions must be satisfied: 1. \(\mathbf{U} \neq \mathbf{0}\) and \(\mathbf{V} \neq \mathbf{0}\) (already given) 2. The two direction vectors should be proportional to each other, which means that there is a scalar value \(k\) such that: \(\mathbf{U} = k\mathbf{V}\). 3. Any point on the first line must also be a point on the second line. That means that both points, \(\mathbf{X}_0\) and \(\mathbf{X}_1\), are part of the same line, so there is a value of \(t\) and \(s\) such that: \(\mathbf{X}_0 + t\mathbf{U} = \mathbf{X}_1 + s\mathbf{V}\).

Express the condition to complete the sentence

Considering all three conditions, we can complete the sentence as follows: The equations $$ \mathbf{X}=\mathbf{X}_{0}+t \mathbf{U}, \quad-\infty<t<\infty $$ and $$ \mathbf{X}=\mathbf{X}_{1}+s \mathbf{V}, \quad-\infty<s<\infty $$ represent the same line in \(\mathbb{R}^{n}\) if and only if their direction vectors are proportional, that is \(\mathbf{U} = k\mathbf{V}\) for some scalar \(k\), and there exists some values of \(t\) and \(s\) such that \(\mathbf{X}_0 + t\mathbf{U} = \mathbf{X}_1 + s\mathbf{V}\).

Key concepts

Vector Equations

Vector equations, such as \begin{align*}\mathbf{X} &= \mathbf{X}_{0} + t \mathbf{U}, \end{align*}are fundamental in real analysis for representing lines, planes, and more complex geometric objects in spaces like \(\mathbb{R}^{n}\). The variable \(\mathbf{X}\) can be thought of as a moving point in the space, \(\mathbf{X}_{0}\) is a given point through which the line passes, and \(\mathbf{U}\) serves as the direction vector that determines the line's orientation. The scalar \(\mathbf{t}\) allows \(\mathbf{X}\) to 'slide' along the line in both directions, painting the entirety of the line as \(\mathbf{t}\) takes on all real values. Understanding vector equations is essential for describing the behavior of lines and various geometrical constructs in higher dimensions.

Direction Vectors

Direction vectors are pivotal in determining the orientation of a line or plane in multi-dimensional spaces. A direction vector like \(\mathbf{U}\) in the equation \(\mathbf{X} = \mathbf{X}_{0} + t \mathbf{U}\) dictates the line's trajectory in space. It's a vector that 'points' along the line, indicating where the line proceeds from the starting point \(\mathbf{X}_{0}\). It is important to note that a line has an infinite number of direction vectors, all of which are scalars multiples of each other (they are collinear to one another). Hence, the particular choice of direction vector does not change the line it represents, as long as it is not the zero vector \(\mathbf{0}\) – because a direction vector cannot signify a direction if it doesn't have one itself!

Proportional Vectors

Proportional vectors, which are also referred to as collinear or parallel vectors, come into play when comparing the orientation of two lines. In the context of our textbook exercise, for two lines to be identical in \(\mathbb{R}^{n}\), the direction vectors \(\mathbf{U}\) and \(\mathbf{V}\) must be proportional. This means that one vector is a scalar multiple of the other: \(\mathbf{U} = k\mathbf{V}\) for some non-zero scalar \(\mathbf{k}\). When vectors are proportional, they share the same (or exactly opposite) direction, which implies they lay on the same line or plane in space. Recognizing proportionality between vectors is a key skill in determining the relationships between lines, planes, and their respective orientations.

Scalar Multiplication

Scalar multiplication is an operation that involves a vector \(\mathbf{V}\) and a scalar \(\mathbf{k}\), resulting in a new vector \(\mathbf{kV}\). In essence, scalar multiplication stretches or contracts the original vector by the magnitude of the scalar and may reverse its direction if the scalar is negative. For instance, \(\mathbf{U} = k\mathbf{V}\) illustrates scalar multiplication where the direction vector \(\mathbf{U}\) is \(\mathbf{k}\) times the vector \(\mathbf{V}\). Scalar multiplication is vital for problems involving vector equations as it allows for adjusting the magnitude and direction of vectors to represent geometrical entities accurately. A deep understanding of this operation is crucial when dealing with geometric transformations and vector spaces in real analysis.