Question

Show that if a line passes through the origin, the vectors of points on the line are all scalar multiples of some fixed nonzero vector.

Short answer

Vectors on the line passing through origin are indeed scalar multiples of a fixed vector.

Step-by-step solution

Understand the Condition of the Line Passing Through Origin

When a line passes through the origin, any point \((x, y)\) on this line can be represented as a vector \(\mathbf{r} = \langle x, y \rangle\), where the line equation is of the form \(y = mx\), with \(m\) being the slope. The fact that the line passes through the origin implies the general equation can be written as \( \mathbf{r} = t \mathbf{v} \), where \(\mathbf{v} = \langle a, b \rangle\).

Define a Scalar Multiple Expression

Since we want all points on the line to be scalar multiples of some vector \(\mathbf{v}\), express the position vector \((x, y)\) as a scalar multiple: \(\mathbf{r} = t\mathbf{v}\), where \(t\) is a scalar and \(\mathbf{v} = \langle a, b \rangle \).

Find Connection Between Line Equation and Scalar

Using the line equation \(y = mx\), substitute \((x, y) = (ta, tb)\) such that \( ta = x \) and \( tb = y \), which satisfies the equation if \( b = ma \). This shows that the vectors on the line are indeed \( \mathbf{r} = t \mathbf{v} \).

Conclude the Solution

The conclusion here is that any point on the line can be reached by scaling \(\mathbf{v}\), proving that every vector on the line is a scalar multiple of this fixed vector \(\mathbf{v} = \langle a, ma \rangle \), confirming our initial assumption.

Key concepts

Scalar Multiplication

In the world of vectors, scalar multiplication is a fundamental operation. Imagine you have a vector and you want to change its size without altering its direction. This is achieved through scalar multiplication.

When you multiply a vector by a scalar, you're essentially stretching or shrinking the vector. If the scalar is positive, the vector retains its direction; if it's negative, the direction reverses. For any vector \( \mathbf{v} = \langle a, b \rangle \), multiplying by a scalar \( t \) results in a new vector \( t \mathbf{v} = \langle ta, tb \rangle \).

This is crucial in understanding lines through the origin, where each point on the line can be generated through scalar multiplication of a fixed vector.

Line Equation

The equation of a line in vector terms often involves finding all points that lie on it. For lines passing through the origin, the equation takes a simplified form.

A line through the origin in a 2D vector space can be represented by the equation \( y = mx \), where \( m \) is the slope. This expression translates perfectly into vector terms as \( \mathbf{r} = t \mathbf{v} \), where \( \mathbf{v} \) is a fixed vector.

This means any point on the line is simply a scalar multiple of a specific vector, making its representation straightforward.

Origin in Vector Space

The origin, denoted as \( (0, 0) \) in a two-dimensional space, holds special significance in vector mathematics. It serves as the starting point or the anchor for vectors.

When discussing lines that pass through the origin, it implies that the line stretches infinitely in both directions from this central point. Any vector from this origin represents a direction and option for scaling, offering limitless possibilities through scalar multiplication.

In essence, lines through the origin simplify mathematical expressions, often allowing vectors to be defined solely by scalar relationships.

Vector Representation

Vectors offer a powerful way to represent points and directions in space. Each vector can be visualized as an arrow pointing from the origin to a specific point.

In the context of a line through the origin, every point on the line is represented as a vector \( \mathbf{r} = \langle x, y \rangle \). This vector can also be expressed as a scalar multiple of a fixed vector \( \mathbf{v} \).

This representation simplifies the concept immensely. Instead of thinking of each point separately, we think in terms of scaling a single vector to reach any point on the line, showcasing the power and elegance of vector mathematics.