The Triangle Inequality in Vector Addition

Problem: The Triangle Inequality in Vector Addition

In classical mechanics and vector algebra, physical quantities such as displacement, velocity, acceleration, and force are represented as vectors. Let ##\vec{A}## and ##\vec{B}## be two vectors in a multi-dimensional vector space. Prove using vector algebra (specifically the properties of the vector dot product) and geometric principles that the magnitude of their resultant vector ##\vec{R} = \vec{A} + \vec{B}## satisfies the triangle inequality:

Provide a complete, mathematically rigorous, step-by-step derivation and analyze the physical conditions under which the equalities hold.

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Worked Solution & Step-by-Step Explanation

To establish this fundamental inequality, we will perform an algebraic proof using the properties of the vector dot product, followed by a geometric analysis of the vector triangle.

Step 1: Algebraic Formulation via the Vector Dot Product

Let the resultant vector be defined as:

Using the definition of the dot product, where ##\theta## is the angle between ##\vec{A}## and ##\vec{B}## such that ##0 \le \theta \le \pi##:

Substituting this back into the expression for the squared magnitude of the resultant:

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Step 2: Proving the Upper Bound (The Right-Hand Inequality)

The trigonometric cosine function is bounded for all real angles ##\theta## by the interval:

To find the maximum possible magnitude of the resultant vector, we substitute the upper bound ##\cos\theta \le 1## into the magnitude equation:

Recognizing the right side of the inequality as a perfect square trinomial:

Substituting back ##\vec{R} = \vec{A} + \vec{B}##:

This completes the proof of the upper bound. Physically, this states that the magnitude of the displacement resulting from two successive paths cannot exceed the sum of the individual path distances.

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Step 3: Proving the Lower Bound (The Left-Hand Inequality)

To determine the minimum possible magnitude of the resultant vector, we apply the lower bound of the cosine function, ##\cos\theta \ge -1##, to the magnitude equation:

We express the right side as a perfect square:

Taking the square root of both sides of the inequality yields:

Substituting back ##\vec{R} = \vec{A} + \vec{B}##:

This completes the proof of the lower bound. Physically, this indicates that the magnitude of the combined vector cannot be less than the net difference between the magnitudes of the individual components.

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Step 4: Combining the Bounds

By combining the inequality statements proved in Step 2 and Step 3, we arrive at the complete double inequality:

This mathematical statement is known as the Triangle Inequality in vector spaces.

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Structural Reference Tables

The following tables outline the variables, parameters, and key boundary conditions that dictate the behavior of vector addition.

Table 1: Definition of Vector Parameters

The absolute scalar length of each vector (always ##\ge 0##)

Resultant vector

##\vec{R} = \vec{A} + \vec{B}##

The net vector sum representing the combined physical effect

Orientation angle

##\theta##

The angle between the direction of ##\vec{A}## and ##\vec{B}## when placed tail-to-tail

Table 2: Analysis of Critical Boundary Conditions

Physical Alignment Description

##0## (or ##0^\circ##)

##1##

Collinear vectors acting in the same direction (maximum possible resultant).

##\dfrac{\pi}{2}## (or ##90^\circ##)

##0##

Orthogonal vectors; magnitude calculated via the Pythagorean theorem.

##\pi## (or ##180^\circ##)

##-1##

Collinear vectors acting in directly opposing directions (minimum possible resultant).

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Geometric Interpretation

The Upper Bound: In Euclidean geometry, the length of any single side of a triangle must be strictly less than the sum of the lengths of the other two sides. If the vectors are collinear and point in the same direction (##\theta = 0##), the triangle collapses into a straight line segment, and the length of the path equals the algebraic sum: ##	\vec{R}	=	\vec{A}	+	\vec{B}	##.
The Lower Bound: Similarly, the length of any side of a triangle must be greater than or equal to the absolute difference of the other two sides. If the vectors are collinear and point in opposite directions (##\theta = \pi##), the triangle collapses, and the net displacement is the absolute difference: ##	\vec{R}	=		\vec{A}	-	\vec{B}

This completes the logical, algebraic, and geometric proof of the triangle inequality for vector addition.