How to Add Vectors: Methods and Examples Explained

A vector is a mathematical quantity characterized by both magnitude and direction, distinguished from scalars which have only magnitude. In three-dimensional Euclidean space, vectors are fundamental objects for describing physical and geometric phenomena such as displacement, velocity, and force. Operations on vectors, particularly addition, are governed by precise algebraic and geometric laws, which must be rigorously stated and derived to ensure mathematical consistency in both pure and applied contexts.

Addition of Vectors in Cartesian and Component Form

Let $\vec{A}$ and $\vec{B}$ be arbitrary vectors in three-dimensional Cartesian space. These vectors may be expressed in terms of their components along the standard basis vectors $\hat{\imath}$, $\hat{\jmath}$, and $\hat{k}$ as follows: \[ \vec{A} = A_x \hat{\imath} + A_y \hat{\jmath} + A_z \hat{k} \] \[ \vec{B} = B_x \hat{\imath} + B_y \hat{\jmath} + B_z \hat{k} \] where $A_x, A_y, A_z, B_x, B_y, B_z \in \mathbb{R}$. The sum $\vec{R}$ of vectors $\vec{A}$ and $\vec{B}$ is defined component-wise: \[ \vec{R} = \vec{A} + \vec{B} = (A_x + B_x) \hat{\imath} + (A_y + B_y) \hat{\jmath} + (A_z + B_z) \hat{k} \]

If $\vec{A}$ and $\vec{B}$ are vectors in two-dimensional space, they are described as \[ \vec{A} = A_x \hat{\imath} + A_y \hat{\jmath} \] \[ \vec{B} = B_x \hat{\imath} + B_y \hat{\jmath} \] and their sum is \[ \vec{A} + \vec{B} = (A_x + B_x)\hat{\imath} + (A_y + B_y)\hat{\jmath} \]

The magnitude $|\vec{R}|$ and direction $\theta$ of the resultant vector $\vec{R} = R_x \hat{\imath} + R_y \hat{\jmath}$ are given by: \[ |\vec{R}| = \sqrt{R_x^2 + R_y^2} \] \[ \theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) \] where $R_x = A_x + B_x$ and $R_y = A_y + B_y$.

Geometrical Interpretation of Vector Addition: Triangle Law

Given two vectors $\vec{P}$ and $\vec{Q}$, the triangle law states that if $\vec{P}$ is represented as a directed line segment, and $\vec{Q}$ is represented such that its tail coincides with the head of $\vec{P}$, then the vector sum $\vec{P} + \vec{Q}$ is given by the directed segment drawn from the tail of $\vec{P}$ to the head of $\vec{Q}$. Explicitly, \[ \vec{P} + \vec{Q} = \vec{R} \] where $\vec{R}$ completes the triangle.

The construction proceeds as follows. Set point $O$ as the origin. Represent $\vec{P}$ by the segment $OA$. From point $A$, draw the segment $AB$ such that $AB$ is equal in magnitude and direction to $\vec{Q}$. The vector $OB$ is, by definition, the sum $\vec{P} + \vec{Q}$.

Geometrical Interpretation of Vector Addition: Parallelogram Law

Given vectors $\vec{a}$ and $\vec{b}$ originating from the same point $O$, the parallelogram law states that if a parallelogram is constructed with $\vec{a}$ and $\vec{b}$ as adjacent sides from $O$, then the diagonal passing through $O$ represents the sum $\vec{a} + \vec{b}$ in both magnitude and direction. Algebraically, \[ \vec{a} + \vec{b} = \text{diagonal } OC \text{ of parallelogram } OABC \] where $OA = \vec{a}$ and $OB = \vec{b}$.

Both triangle law and parallelogram law yield the same resultant, establishing their equivalence for binary vector addition. For a deeper exploration of vector operations, refer to Vector Algebra.

Magnitude of Resultant Vector Using Parallelogram Law

Let vectors $\vec{a}$ and $\vec{b}$ originate from a common point and let the angle between them be $\theta$, where $0 \leq \theta \leq \pi$. The magnitude of their sum, denoted by $|\vec{a} + \vec{b}|$, is obtained by employing the law of cosines: \[ |\vec{a} + \vec{b}| = \sqrt{a^2 + b^2 + 2ab\cos\theta} \] where $a=|\vec{a}|$, $b=|\vec{b}|$, and $\theta$ is the angle from $\vec{a}$ to $\vec{b}$ measured at their common origin.

This result is derived as follows. Let $|\vec{a}| = a$, $|\vec{b}| = b$, and let $\vec{a}$ and $\vec{b}$ be placed tail-to-tail. The resultant vector $\vec{R} = \vec{a} + \vec{b}$. Then, \[ |\vec{R}|^2 = (\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b}) \] Expanding, \[ |\vec{R}|^2 = \vec{a} \cdot \vec{a} + 2\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} \] \[ |\vec{R}|^2 = a^2 + 2ab\cos\theta + b^2 \] Taking square roots, \[ |\vec{R}| = \sqrt{a^2 + b^2 + 2ab\cos\theta} \]

Algebraic Addition of Vectors Using Ordered Pairs

In $\mathbb{R}^n$, if vectors $\vec{P} = (p_1, p_2, \ldots, p_n)$ and $\vec{Q} = (q_1, q_2, \ldots, q_n)$ are represented as ordered $n$-tuples, the addition is defined component-wise: \[ \vec{P} + \vec{Q} = (p_1 + q_1, p_2 + q_2, \ldots, p_n + q_n) \] In two dimensions: \[ \vec{P} = (p_1, p_2),\quad \vec{Q} = (q_1, q_2) \] \[ \vec{P} + \vec{Q} = (p_1 + q_1,\, p_2 + q_2) \]

Properties of Vector Addition (Algebraic Structure)

Vector addition in $\mathbb{R}^n$ possesses several algebraic properties critical for further vector algebra:

Commutativity: For any vectors $\vec{A}$ and $\vec{B}$, \[ \vec{A} + \vec{B} = \vec{B} + \vec{A} \]

Associativity: For any vectors $\vec{A}$, $\vec{B}$, and $\vec{C}$, \[ (\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C}) \]

Existence of Additive Identity: There exists a unique zero vector $\vec{0}$ such that for all $\vec{A}$, \[ \vec{A} + \vec{0} = \vec{A} \]

Existence of Additive Inverse: For every vector $\vec{A}$, there exists a vector $-\vec{A}$ such that \[ \vec{A} + (-\vec{A}) = \vec{0} \]

Worked Examples on Addition of Vectors

Example 1: Two vectors $\vec{A} = (3,4)$ and $\vec{B} = (2,6)$ are given. Compute $\vec{A} + \vec{B}$.

Substitution: \[ \vec{A} + \vec{B} = (3+2, 4+6) = (5, 10) \]

Final result: $\vec{A} + \vec{B} = (5, 10)$

Example 2: Given vectors $\vec{A} = (2, 3)$ and $\vec{B} = (2, -2)$, determine the magnitude $|\vec{C}|$ and direction $\phi$ of their sum $\vec{C} = \vec{A} + \vec{B}$.

Substitution: \[ \vec{C} = \vec{A} + \vec{B} = (2+2, 3+(-2)) = (4, 1) \]

Magnitude calculation: \[ |\vec{C}| = \sqrt{4^2 + 1^2} = \sqrt{16+1} = \sqrt{17} \]

Angle calculation: \[ \phi = \tan^{-1}\left(\frac{1}{4}\right) \]

Final result: $|\vec{C}| = \sqrt{17} \approx 4.123$, $\phi \approx 14.04^{\circ}$

Example 3: Given $\vec{a} = (1, -1)$ and $\vec{b} = (2, 1)$, obtain the unit vector in the direction of $\vec{a} + \vec{b}$.

Substitution: \[ \vec{a} + \vec{b} = (1+2, -1+1) = (3, 0) \]

Magnitude calculation: \[ |\vec{a} + \vec{b}| = \sqrt{3^2 + 0^2} = 3 \]

Unit vector: \[ \frac{\vec{a} + \vec{b}}{|\vec{a} + \vec{b}|} = \frac{(3, 0)}{3} = (1, 0) \]

Final result: The required unit vector is $(1, 0)$

Generalisation: Polygon Law of Vector Addition

If $n$ vectors $\vec{A}_1, \vec{A}_2, \ldots, \vec{A}_n$ are represented by contiguous sides of a polygon arranged in order, then the closing side taken in reverse order represents the vector sum \[ \vec{A}_1 + \vec{A}_2 + \cdots + \vec{A}_n \] This principle extends the triangle and parallelogram rules for arbitrary finite collections of vectors. For focused results pertaining to summing three vectors, consult Addition Of Three Vectors.

Conditions and Limitations in Addition of Vectors

Only vectors of the same type and defined in the same dimensional space may be added. For example, physical vectors such as displacement can only be added with other displacement vectors. Addition of vectors and scalars is undefined. Additionally, the vectors must be referenced relative to compatible coordinate axes or systems.

Frequently Queried Aspects in Vector Addition

Two vectors are said to be additive inverses if their sum yields the zero vector. The negative of a vector retains the same magnitude as the original but is directed oppositely. Additive identity in vector space corresponds to the zero vector whose magnitude is zero and direction is indeterminate. For detailed exploration of operations involving negative vectors, see Subtraction Of Vectors.

Summary of Main Results on Addition of Vectors

Addition of vectors is rigorously defined both algebraically via componentwise summation and geometrically through the triangle and parallelogram laws. The results fundamentally rely on vector space axioms, including commutativity, associativity, existence of additive identity, and additive inverse. The magnitude of the sum of two vectors can be calculated using the formula involving the cosine of the included angle. For advanced study including scalar multiplication, vector products, and properties in higher-dimensional spaces, refer to relevant resources such as Vector Triple Product and Scalar Product Of Vectors.