Question

Prove that \(\bar{x}\) and \(\bar{y}\) are parallel iff $$ \frac{x_{1}}{y_{1}}=\frac{x_{2}}{y_{2}}=\cdots=\frac{x_{n}}{y_{n}}=c \quad\left(c \in E^{1}\right) $$ where " \(x_{k} / y_{k}=c "\) is to be replaced by " \(x_{k}=0\) " if \(y_{k}=0\).

Short answer

Vectors are parallel if \( \frac{x_k}{y_k} = c \) for each component, or \( x_k = 0 \) if \( y_k = 0 \).

Step-by-step solution

Understand Vectors and Parallelism

Two vectors \( \bar{x} \) and \( \bar{y} \) are parallel if one can be expressed as a scalar multiple of the other. This means \( \bar{x} = c \bar{y} \) for some scalar \( c \). In component form, this can be expressed as \( x_k = c y_k \) for each \( k \).

Express Respective Components Using Scalar Multiplication

If \( \bar{x} = c \bar{y} \), then each component \( x_k = c y_k \). Dividing both sides by \( y_k \) (assuming \( y_k eq 0 \)), we get \( \frac{x_k}{y_k} = c \). If \( y_k = 0 \), then \( x_k \) must also be zero for the equality to hold, expressed as \( x_k = 0 \).

Verify Condition for All Components

Ensure that each component conforms to the condition \( \frac{x_k}{y_k} = c \) or, when \( y_k = 0 \), that \( x_k = 0 \). This confirms that every component ratio is equal to the scalar \( c \), demonstrating parallelism across all dimensions.

Prove the 'if' Part of the Statement

Assume \( \frac{x_{1}}{y_{1}} = \frac{x_{2}}{y_{2}} = \cdots = \frac{x_{n}}{y_{n}} = c \). Then for each component, \( x_k = c y_k \). Thus, \( \bar{x} = c \bar{y} \), proving \( \bar{x} \) and \( \bar{y} \) are parallel.

Prove the 'only if' Part of the Statement

Assume \( \bar{x} \) and \( \bar{y} \) are parallel, so \( \bar{x} = c \bar{y} \). For each component, this gives \( \frac{x_k}{y_k} = c \) if \( y_k eq 0 \), or \( x_k = 0 \) if \( y_k = 0 \). This satisfies the condition \( \frac{x_1}{y_1} = \frac{x_2}{y_2} = \cdots = \frac{x_n}{y_n} = c \).

Key concepts

Vectors

Vectors are a fundamental part of mathematics and physics used to describe quantities with both magnitude and direction. When you think of a vector, envision an arrow: the length represents the magnitude, and the direction in which it points demonstrates its orientation.

In mathematical terms, a vector in \mathbb{R}^n is typically represented as \(\overline{x} = (x_1, x_2, ..., x_n)\). This means that a vector can have multiple components or elements, each representing different axes in an n-dimensional space.

Vectors are often used in various applications such as physics, engineering, and computer science to represent forces, velocities, and even positions. Understanding vectors' nature is crucial for topics like vector parallelism, where relationships between vectors are essential.

Scalar Multiplication

Scalar multiplication is a straightforward yet vital operation in vector algebra. It involves multiplying a vector by a scalar (a real number). Essentially, scalar multiplication scales the vector, affecting its magnitude but not its direction.

When a scalar \(c\) multiplies a vector \(\overline{x} = (x_1, x_2, ..., x_n)\), each component of the vector is scaled by \(c\). The result is expressed as \(c \overline{x} = (c x_1, c x_2, ..., c x_n)\).

Scalar multiplication is key in determining parallelism between vectors. If one vector is a scalar multiple of another, it signifies they share the same direction, and thus, they are parallel. This is fundamental in many computational problems and graphical interpretations of vectors.

Component Form

Vectors can be broken down into their component form, which helps to easily see and compare each part of the vector. Component form involves expressing the vector \(\overline{x}\) as \( (x_1, x_2, ..., x_n)\), where each component \(x_k\) corresponds to the vector's projection along an axis in n-dimensional space.

Understanding vectors in component form simplifies processes like addition, subtraction, and scalar multiplication because these operations can be performed separately on each component. For example, adding vectors \(\overline{x}\) and \(\overline{y}\) can be done directly using their components: \( (x_1 + y_1, x_2 + y_2, ..., x_n + y_n)\).

Component form is also useful for verifying vector parallelism, as it allows the comparison of corresponding elements to see if a constant ratio exists among them.

Parallel Vectors

Parallel vectors are vectors that share the same or exact opposite directions. In mathematical terms, two vectors \(\overline{x}\) and \(\overline{y}\) are parallel if there exists a scalar \(c\) such that \(\overline{x} = c\overline{y}\).

This means that for each component of the vectors, the ratio \( \frac{x_k}{y_k} = c\) must hold true, provided \(y_k eq 0\). If any \(y_k\) is zero, then the corresponding \(x_k\) should also be zero. This ensures that the magnitude of one vector can be transformed into the magnitude of the other by scaling, maintaining the same direction.

Recognizing parallel vectors is essential in solving geometric problems and understanding physical phenomena, such as when two forces are acting in the same line, leading to zero net force if they are opposite but equal in magnitude.