Question

Given that \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}=\overrightarrow{\mathbf{C}}\) and that \(A+B=C\), how are \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) oriented relative to one another?

Short answer

\(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) are collinear and in the same direction.

Step-by-step solution

Understand the Vector Equation

We are given that \( \overrightarrow{\mathbf{A}} + \overrightarrow{\mathbf{B}} = \overrightarrow{\mathbf{C}} \). This means that the vector sum of \( \overrightarrow{\mathbf{A}} \) and \( \overrightarrow{\mathbf{B}} \) results in the vector \( \overrightarrow{\mathbf{C}} \).

Analyze the Magnitudes

The problem also states that the magnitudes of the vectors satisfy the equation \( A + B = C \), where \( A = |\overrightarrow{\mathbf{A}}| \), \( B = |\overrightarrow{\mathbf{B}}| \), and \( C = |\overrightarrow{\mathbf{C}}| \).

Apply the Triangle Inequality

For two vectors \( \overrightarrow{\mathbf{A}} \) and \( \overrightarrow{\mathbf{B}} \), the triangle inequality states that \( A + B \geq C \). However, we have \( A + B = C \), which indicates that the vectors must be collinear and in the same direction.

Determine the Orientation

Since \( A + B = C \) exactly meets the case of equality in the triangle inequality, this implies \( \overrightarrow{\mathbf{A}} \) and \( \overrightarrow{\mathbf{B}} \) are oriented in the same direction on a straight line.

Key concepts

Vector Magnitude

The concept of vector magnitude is vital in understanding vectors and their properties. The magnitude of a vector, often referred to as its length or norm, is the distance from the origin to the point represented by the vector. It acts like a measure of how long the vector is.

• For any vector \( \overrightarrow{\mathbf{A}} \), its magnitude is denoted as \( |\overrightarrow{\mathbf{A}}| \), and it's always a non-negative number.
• In a two-dimensional space, if \( \overrightarrow{\mathbf{A}} = (x, y) \), then its magnitude is calculated as \( |\overrightarrow{\mathbf{A}}| = \sqrt{x^2 + y^2} \).
• In a three-dimensional space, for \( \overrightarrow{\mathbf{A}} = (x, y, z) \), its magnitude becomes \( |\overrightarrow{\mathbf{A}}| = \sqrt{x^2 + y^2 + z^2} \).

Understanding vector magnitude helps analyze vector equations and conditions, such as when determining if vectors satisfy certain inequalities or conditions, like in the current exercise.

Triangle Inequality

The triangle inequality is a fundamental principle when dealing with vectors. It states that the sum of the magnitudes of two vectors is always greater than or equal to the magnitude of their sum. Formally, for vectors \( \overrightarrow{\mathbf{A}} \) and \( \overrightarrow{\mathbf{B}} \), it can be expressed as \( A + B \geq C \), where:

• \( A = |\overrightarrow{\mathbf{A}}| \) is the magnitude of vector \( \overrightarrow{\mathbf{A}} \).
• \( B = |\overrightarrow{\mathbf{B}}| \) is the magnitude of vector \( \overrightarrow{\mathbf{B}} \).
• \( C = |\overrightarrow{\mathbf{C}}| \) where \( \overrightarrow{\mathbf{C}} = \overrightarrow{\mathbf{A}} + \overrightarrow{\mathbf{B}} \).

In this exercise, the condition \( A + B = C \) indicates that the two vectors are collinear and point in the same direction. This exact equality signifies that the path traced by these vectors forms a straight line, which is a special case in the triangle inequality.

Collinear Vectors

Collinear vectors are vectors that lie on the same straight line, meaning they have the same or exact opposite direction. When vectors are collinear, there exists a scalar factor that can multiply one vector to turn it into the other. This is a key concept for identifying the orientation of vectors in terms of parallel alignment.

• Vectors \( \overrightarrow{\mathbf{A}} \) and \( \overrightarrow{\mathbf{B}} \) are collinear if there exists a scalar \( k \) such that \( \overrightarrow{\mathbf{A}} = k \cdot \overrightarrow{\mathbf{B}} \).
• If \( k > 0 \), the vectors point in the same direction.
• If \( k < 0 \), the vectors point in opposite directions.

In scenarios like the given exercise, where the magnitudes satisfy \( A + B = C \) and the vector sum aligns perfectly, this implies that \( \overrightarrow{\mathbf{A}} \) and \( \overrightarrow{\mathbf{B}} \) are not only collinear but also in the same direction, further indicating that the sum of the vectors extends to form a straight line.