How to Subtract Vectors: A Simple Student Guide

The subtraction of vectors is the operation that, given two vectors of the same kind, produces a third vector representing their difference, defined through addition with the negative of the second vector.

Definition and Notation of Vector Subtraction

Definition: If $\vec{a}$ and $\vec{b}$ are vectors of the same physical quantity, their difference is denoted as $\vec{a} - \vec{b}$ and defined by $\vec{a} - \vec{b} = \vec{a} + (-\vec{b})$, where $-\vec{b}$ is the additive inverse of $\vec{b}$, possessing identical magnitude and opposite direction.

Geometric Interpretation Using the Parallelogram Law

To subtract $\vec{b}$ from $\vec{a}$ by the parallelogram law, represent both vectors from a common initial point. The vector $-\vec{b}$ is constructed by reversing the direction of $\vec{b}$, and the resultant $\vec{a} + (-\vec{b})$ forms the diagonal of the parallelogram built on $\vec{a}$ and $-\vec{b}$ sharing the same origin.

Graphical Construction via the Triangle Law

According to the triangle law, place $\vec{b}$ and $-\vec{b}$ tail-to-tail. Draw $\vec{a}$ from the same origin as $\vec{b}$; the vector from the tip of $\vec{b}$ to the tip of $\vec{a}$ (with both vectors coinitial) represents $\vec{a} - \vec{b}$.

Componentwise Formula for Vector Subtraction

If $\vec{a} = \langle a_1, a_2, a_3 \rangle$ and $\vec{b} = \langle b_1, b_2, b_3 \rangle$, then $\vec{a} - \vec{b} = \langle a_1-b_1, a_2-b_2, a_3-b_3 \rangle$. Each component is subtracted independently, which is a direct result of the definition of vector addition and scalar multiplication.

Essential Properties of the Subtraction Operation

The following conceptual points should be noted regarding vector subtraction:

• Only defined between vectors of identical physical type
• Not commutative; $\vec{a} - \vec{b} \neq \vec{b} - \vec{a}$ in general
• Not associative; $(\vec{a} - \vec{b}) - \vec{c} \neq \vec{a} - (\vec{b} - \vec{c})$
• $\vec{a} - \vec{a} = \vec{0}$ always holds
• Magnitude depends on the angle and moduli

For a detailed study of related ideas, refer to Addition Of Vectors and Vector Algebra.

Magnitude of the Difference Vector and Algebraic Relations

The square of the magnitude of $\vec{a} - \vec{b}$ follows from the law of cosines and properties of the dot product: $\|\vec{a} - \vec{b}\|^2 = \|\vec{a}\|^2 + \|\vec{b}\|^2 - 2\,\vec{a} \cdot \vec{b}$. This result is essential for quantitative questions involving the angle between vectors.

Illustrative Calculations Involving Vector Subtraction

Example: Given $\vec{a} = \langle 4, -2, 3 \rangle$, $\vec{b} = \langle 1, -2, 5 \rangle$, compute $\vec{a} - \vec{b}$.

Subtract components: $4-1=3$, $-2-(-2)=0$, $3-5=-2$. Thus, $\vec{a} - \vec{b} = \langle 3, 0, -2 \rangle$.

Example: For $\vec{a} = \langle 3, 5 \rangle$, $\vec{b} = \langle 2, 6 \rangle$, evaluate $\vec{a} - \vec{b}$.

Compute: $3-2=1$, $5-6=-1$. The result is $\langle 1, -1 \rangle$.

Example: If $\vec{a}$ and $\vec{b}$ are unit vectors with angle $\theta$ between them, determine $\|\vec{a} - \vec{b}\|$ in terms of $\theta$.

$\|\vec{a} - \vec{b}\|^2 = 1 + 1 - 2\cos\theta = 2(1-\cos\theta) = 4\sin^2\left(\dfrac{\theta}{2}\right)$, so $\|\vec{a} - \vec{b}\| = 2\sin \left(\dfrac{\theta}{2}\right)$.

Example: If $\|\vec{a} + \vec{b}\| = \|\vec{a} - \vec{b}\|$, prove the vectors are orthogonal.

Square both sides: $\|\vec{a} + \vec{b}\|^2 = \|\vec{a} - \vec{b}\|^2$. So $2\vec{a} \cdot \vec{b} = -2\vec{a} \cdot \vec{b}$, implying $\vec{a} \cdot \vec{b} = 0$. Therefore, $\vec{a}$ and $\vec{b}$ are perpendicular.

Frequent Misconceptions in Vector Subtraction

Common Error: Attempting to subtract a scalar from a vector or subtract vectors of different types is undefined; subtraction applies only to vectors of the same physical nature and dimensionality.

For further operations on vectors, see Scalar Product Of Vectors and Vector Triple Product for related algebraic structure.

Componentwise subtraction, geometric visualization, and the non-commutative character of subtraction constitute essential points for problem-solving in vector algebra and System Of Linear Equations contexts.