How to Add Three Vectors: Step-by-Step Guide

The addition of three vectors involves combining their magnitudes and directions according to the axioms of vector addition. This process is essential in advanced vector algebra and underpins the analysis of physical quantities such as force, velocity, and displacement. The addition may be performed using geometric or analytical methods, subject to the representation of the vectors involved.

Addition of Three Vectors Using Analytical Methods

Let $\vec{A}$, $\vec{B}$, and $\vec{C}$ be three vectors in three-dimensional space. The resultant vector, denoted as $\vec{R}$, is defined by the equation \[ \vec{R} = \vec{A} + \vec{B} + \vec{C} \] where addition is performed component-wise.

Express $\vec{A}$, $\vec{B}$, and $\vec{C}$ with respect to the standard basis vectors: \[ \vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k} \] \[ \vec{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k} \] \[ \vec{C} = C_x \hat{i} + C_y \hat{j} + C_z \hat{k} \] where $A_x, A_y, A_z$, etc., denote the respective components along the $x$-, $y$-, and $z$-axes.

The sum of the three vectors is then calculated component-wise: \[ \vec{R} = (A_x+B_x+C_x) \hat{i} + (A_y+B_y+C_y) \hat{j} + (A_z+B_z+C_z) \hat{k} \]

Denoting the resultant components as $R_x = A_x+B_x+C_x$, $R_y = A_y+B_y+C_y$, $R_z = A_z+B_z+C_z$, the magnitude of the resultant vector is given by \[ |\vec{R}| = \sqrt{R_x^2 + R_y^2 + R_z^2} \]

For the case that vectors lie in a plane (planar vectors), the $z$-components vanish, and the resultant is simply \[ |\vec{R}| = \sqrt{R_x^2 + R_y^2} \]

Addition of Three Vectors by Successive Geometric Laws

If magnitudes and the angles between the vectors are specified, but not their components, the resultant is determined through iterative application of the triangle law of vector addition. For two arbitrary vectors $\vec{P}$ and $\vec{Q}$ separated by an angle $\theta$, the triangle law states: \[ |\vec{P}+\vec{Q}| = \sqrt{P^2 + Q^2 + 2PQ \cos\theta} \] where $P = |\vec{P}|$, $Q = |\vec{Q}|$, and $\theta$ is the angle from $\vec{P}$ to $\vec{Q}$.

To add three vectors $\vec{A},\ \vec{B},\ \vec{C}$ represented by magnitudes and angles:

First, find $\vec{R}_1=\vec{A}+\vec{B}$. If the angle between $\vec{A}$ and $\vec{B}$ is $\theta_1$, \[ |\vec{R}_1| = \sqrt{A^2 + B^2 + 2AB\cos\theta_1} \]

Denote $\alpha$ as the angle between $\vec{R}_1$ and $\vec{C}$. The final resultant is computed as \[ |\vec{R}| = \sqrt{|\vec{R}_1|^2 + C^2 + 2|\vec{R}_1|C\cos\alpha} \]

Application of this law requires explicit knowledge of the angles between each step’s resultant and the next vector; geometric construction aids in determining these where not given. For a graphical depiction, refer to the triangle and parallelogram addition laws as detailed in Addition Of Vectors.

Graphical Illustration of Three Vector Addition

The addition of three vectors graphically involves placing the tail of each subsequent vector at the head of the previous one. The resultant $\vec{R}$ is drawn from the tail of the first vector to the head of the last vector. The graphical technique enables visualization of the resultant’s magnitude and direction. Refer to the illustration below:

Detailed Derivation: Magnitude of the Resultant When Three Directions and Angles Are Known

Let three vectors $\vec{A},\ \vec{B},\ \vec{C}$ be such that the angle between $\vec{A}$ and $\vec{B}$ is $\theta_1$, and the angle between $\vec{R}_1 = \vec{A}+\vec{B}$ and $\vec{C}$ is $\theta_2$. To determine the magnitude of the resultant step by step:

First, find $|\vec{R}_1|$: \[ |\vec{R}_1| = \sqrt{A^2 + B^2 + 2AB \cos\theta_1} \]

Next, compute $|\vec{R}|$ where $\vec{R} = \vec{R}_1 + \vec{C}$, and $\theta_2$ is the angle between $\vec{R}_1$ and $\vec{C}$. \[ |\vec{R}| = \sqrt{|\vec{R}_1|^2 + C^2 + 2|\vec{R}_1|C\cos\theta_2} \]

Every intermediate calculation must be performed completely and explicitly in practical problems. For details on parallelogram and triangle laws, see Vector Algebra.

Stepwise Example: Component Method for Three Vectors

Given: $\vec{A} = 3\hat{i} + 2\hat{j} + 6\hat{k}$, $\vec{B} = 5\hat{i} - 3\hat{j} + 4\hat{k}$, $\vec{C} = -1\hat{i} + 7\hat{j} - 2\hat{k}$

Resultant: \[ \vec{R} = (3 + 5 + (-1))\hat{i} + (2 + (-3) + 7)\hat{j} + (6 + 4 + (-2))\hat{k} \]

Calculate each component: \[ \begin{align*} R_x &= 3 + 5 + (-1) = 7 \\ R_y &= 2 + (-3) + 7 = 6 \\ R_z &= 6 + 4 + (-2) = 8 \\ \end{align*} \] So, \[ \vec{R} = 7\hat{i} + 6\hat{j} + 8\hat{k} \]

Magnitude: \[ |\vec{R}| = \sqrt{(7)^2 + (6)^2 + (8)^2} \] \[ |\vec{R}| = \sqrt{49 + 36 + 64} \] \[ |\vec{R}| = \sqrt{149} \] \[ |\vec{R}| = 12.206\ (\text{to three decimal places}) \]

For further vector manipulation techniques, see Scalar Product Of Vectors.

Stepwise Example: Magnitude by Successive Triangle Law

Given: $|\vec{A}|=4$ units, $|\vec{B}|=5$ units, $|\vec{C}|=7$ units. The angle between $\vec{A},\vec{B}$ is $90^\circ$, angle between their resultant $\vec{R}_1$ and $\vec{C}$ is $60^\circ$.

First, apply triangle law to $\vec{A}$ and $\vec{B}$: \[ |\vec{R}_1| = \sqrt{4^2 + 5^2 + 2 \times 4 \times 5 \times \cos 90^\circ} \] Since $\cos 90^\circ = 0$, \[ |\vec{R}_1| = \sqrt{16 + 25} \] \[ |\vec{R}_1| = \sqrt{41} \] \[ |\vec{R}_1| = 6.403\ \text{units} \ (\text{rounded to three decimal places}) \]

Now, add $\vec{R}_1$ and $\vec{C}$ with an angle $60^\circ$ between them: \[ |\vec{R}| = \sqrt{(\sqrt{41})^2 + 7^2 + 2 \times \sqrt{41} \times 7 \times \cos 60^\circ} \] Since $\cos 60^\circ = \frac{1}{2}$, \[ |\vec{R}| = \sqrt{41 + 49 + 2 \times \sqrt{41} \times 7 \times \frac{1}{2}} \] \[ |\vec{R}| = \sqrt{41 + 49 + 7\sqrt{41}} \] \[ |\vec{R}| = \sqrt{90 + 7\sqrt{41}} \] \[ 7\sqrt{41} \approx 7 \times 6.403 = 44.821 \] \[ 90 + 44.821 = 134.821 \] \[ |\vec{R}| = \sqrt{134.821} \] \[ |\vec{R}| = 11.614\ \text{units} \ (\text{rounded to three decimal places}) \]

For advanced study on triple and cross products, refer to Vector Triple Product.

Criteria for Zero Sum of Three Vectors

Three vectors $\vec{A}, \vec{B}, \vec{C}$ sum to zero if and only if $\vec{A} + \vec{B} + \vec{C} = \vec{0}$, which implies that the vectors form the consecutive sides of a closed triangle taken in the same order. Analytically, for vectors $\vec{A},\vec{B},\vec{C}$ to satisfy this, \[ \vec{A} + \vec{B} = -\vec{C} \] The geometric interpretation is that placement of the vectors head-to-tail results in a return to the starting point.

Connections to the geometric interpretation of vectors can be found in Geometry Of Complex Numbers.

Associativity and Commutativity in Three Vector Addition

Vector addition is both associative and commutative: \[ \vec{A} + (\vec{B} + \vec{C}) = (\vec{A} + \vec{B}) + \vec{C} \] \[ \vec{A} + \vec{B} + \vec{C} = \vec{C} + \vec{B} + \vec{A} \] These properties allow the sum of three or more vectors to be performed in any sequence without altering the result.

For graphical understanding of associative addition, review Graphs Of Sine And Cosine Function.