Understanding Graphical Methods for Vector Addition

The graphical methods of vector addition provide a geometric approach to finding the resultant of two or more vectors by representing vectors as directed line segments in the plane. This topic formalizes such methods with mathematical rigor, covering definitions, construction algorithms, and stepwise analytical connections essential for the JEE Main syllabus.

Formal Representation of Vectors for Graphical Addition

A vector is a directed segment characterized by a magnitude and direction. If a vector $\vec{A}$ is represented in a Cartesian coordinate system, it corresponds to an arrow drawn from initial point $O$ to terminal point $P$. The length of the arrow corresponds to $|\vec{A}|$ (the magnitude), and its direction is specified by the angle $\theta$ it makes with a fixed axis, typically the positive $x$-axis. The mathematical representation is $\vec{A} = |\vec{A}|\hat{n}$, where $\hat{n}$ is the unit vector along its direction.

For graphical operations, vectors are drawn to a scale in a coordinate plane, with the direction measured using a protractor, and their lengths measured using a ruler according to a chosen scale.

One-Dimensional Graphical Addition of Collinear Vectors

For vectors lying along a straight line (collinear), addition reduces to algebraic sum with appropriate sign conventions. Consider two vectors $\vec{A}$ and $\vec{B}$ along the $x$-axis with magnitudes $A$ and $B$ and directions indicated by their arrows. Draw both vectors on a number line starting from the same origin or stacked end-to-end maintaining their respective directions.

If $\vec{A}$ is along positive $x$-axis and $\vec{B}$ is along negative $x$-axis, the resultant vector $\vec{R}$ is given algebraically as $\vec{R} = \vec{A} + \vec{B}$, with the magnitude $|A - B|$ and direction matched to the vector having the greater magnitude.

Head-to-Tail Method for Graphical Vector Addition in the Plane

When adding vectors in two dimensions, the head-to-tail method is the primary graphical procedure. Given vectors $\vec{A}$ and $\vec{B}$, proceed as follows:

First, draw $\vec{A}$ to scale in its proper direction from a chosen origin point. The tail of $\vec{A}$ is at the origin, and the head is at its endpoint.

Next, place the tail of $\vec{B}$ at the head of $\vec{A}$, maintaining both its magnitude (by the same scale) and its direction (measured by a protractor relative to the reference axes).

For the resultant vector $\vec{R}$, draw a new vector from the tail of $\vec{A}$ (original starting point) to the head of $\vec{B}$ (final endpoint after chaining the vectors). This vector $\vec{R}$ represents $\vec{A} + \vec{B}$ in both magnitude and direction.

The magnitude of $\vec{R}$ is measured directly by a ruler using the same scale; its direction is measured as the angle it makes with the reference axis using a protractor.

Graphical Addition for Three or More Vectors

Suppose vectors $\vec{A}$, $\vec{B}$, $\vec{C}$, $\ldots$ are to be added graphically. Draw $\vec{A}$ from an initial point. Then by head-to-tail placement, draw $\vec{B}$ starting at the head of $\vec{A}$, $\vec{C}$ at the head of $\vec{B}$, and so on, preserving each vector's magnitude and angle.

The resultant vector $\vec{R} = \vec{A} + \vec{B} + \vec{C} + \ldots$ is represented as the vector drawn from the tail of the first vector to the head of the final vector in the head-to-tail chain.

Analytical Connection: Triangle Law and Parallelogram Law via Graphical Method

The graphical head-to-tail procedure correlates to the Triangle Law of Vector Addition: the sum $\vec{A} + \vec{B}$ is equivalent to the third side of a triangle taken in the same order. If $\vec{A}$ and $\vec{B}$ are co-located at a point, the Parallelogram Law is constructed by drawing both vectors from a common origin, completing a parallelogram, and the diagonal from the origin gives the resultant graphically.

Explicit Geometrical Steps: Measuring Resultant Magnitude and Direction

Given two vectors $\vec{A}$ and $\vec{B}$ with an angle $\theta$ between them, after constructing the head-to-tail diagram, the magnitude of the resultant $\vec{R}$ can be determined analytically as:

$R = \sqrt{A^2 + B^2 + 2AB\cos\theta}$

Graphically, $R$ is measured using a ruler according to the chosen scale. If direction is sought, measure the angle $\alpha$ that $\vec{R}$ makes with $\vec{A}$ using a protractor, or apply trigonometric calculation:

$\tan\alpha = \dfrac{B\sin\theta}{A + B\cos\theta}$

This establishes equivalence between graphical and analytical results.

Graphical Method of Vector Subtraction

To subtract vector $\vec{B}$ from $\vec{A}$, visualize $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$, where $-\vec{B}$ is a vector equal in magnitude to $\vec{B}$ but opposite in direction. The construction proceeds by drawing $\vec{A}$, then $-\vec{B}$ head-to-tail from the head of $\vec{A}$. The resultant vector is drawn from the tail of $\vec{A}$ to the head of $-\vec{B}$.

Worked Example: Addition of Non-Collinear Vectors Graphically

Given: A person walks $6\,\text{m}$ towards east, then $8\,\text{m}$ towards north. Find the resultant displacement graphically and compare with analytical calculation.

Step 1: Choose a scale, e.g., $1\,\text{cm} = 2\,\text{m}$. Draw vector $\vec{A}$: $6\,\text{m}$ east as a $3\,\text{cm}$ line on the $x$-axis.

Step 2: From the head of $\vec{A}$, draw vector $\vec{B}$: $8\,\text{m}$ north as a $4\,\text{cm}$ line parallel to the $y$-axis.

Step 3: Draw the resultant $\vec{R}$ from the tail of $\vec{A}$ (origin) to the head of $\vec{B}$ (end point).

Step 4: Measure $\vec{R}$ with a ruler. Suppose its length is $5\,\text{cm}$. Thus, $|\vec{R}| = 5 \times 2 = 10\,\text{m}$.

Step 5: Measure the angle $\theta$ with respect to east using a protractor; suppose $\theta = 53^\circ$ north of east.

Analytical Check: By Pythagoras' theorem, $R = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\,\text{m}$. $\tan\theta = 8/6 = 4/3 \implies \theta = \arctan(4/3) \approx 53^\circ$.

Worked Example: Vector Subtraction by Graphical Method

Given: Vector $\vec{P}$ is $5\,\text{units}$ along $30^\circ$ north of east. Vector $\vec{Q}$ is $3\,\text{units}$ along $60^\circ$ north of east. Find $\vec{P}-\vec{Q}$ graphically.

Step 1: Draw $\vec{P}$ to scale at $30^\circ$ north of east.

Step 2: Draw $-\vec{Q}$: $3\,\text{units}$ at $60^\circ$ south of west (opposite direction to $\vec{Q}$).

Step 3: Place $-\vec{Q}$ starting at the head of $\vec{P}$ by head-to-tail rule.

Step 4: The resultant vector from tail of $\vec{P}$ to head of $-\vec{Q}$ gives $\vec{P} - \vec{Q}$ in magnitude and direction, to be measured with ruler and protractor.

Commutativity and Associativity of Graphical Addition

The resultant vector obtained by the graphical method is independent of the order in which vectors are added. That is, for two vectors $\vec{A}$ and $\vec{B}$, $\vec{A} + \vec{B} = \vec{B} + \vec{A}$. Similarly, for three vectors $\vec{A}$, $\vec{B}$, $\vec{C}$, $(\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C})$.

Multiplication of a Vector by a Scalar: Graphical Interpretation

When a vector $\vec{A}$ is multiplied by a scalar $k$, its magnitude changes by a factor $|k|$, and its direction remains the same if $k > 0$, or is reversed if $k < 0$. Graphically, draw a new arrow $k$ times longer (or with reversed direction), starting at any origin. This method is essential for rescaling vectors before addition or subtraction.

Resolution of a Vector Into Components: Graphical Construction

A vector $\vec{A}$ can be resolved graphically into two perpendicular components, typically along $x$ and $y$ axes. Given $\vec{A}$ at angle $\theta$ above $x$-axis, draw $\vec{A}$ from the origin. Drop perpendiculars to the $x$ and $y$ axes from the head of $\vec{A}$. The lengths along $x$- and $y$-axes represent $A_x = |\vec{A}|\cos\theta$ and $A_y = |\vec{A}|\sin\theta$ respectively. This construction allows addition of vectors by adding their respective components along each axis, reinforcing the analytical method with a geometric visualisation.

For further study, refer to Vector Algebra and Addition Of Vectors.