3.2 Vector Addition and Subtraction: Graphical Methods

Vector addition and subtraction are core skills in physics because forces, velocities, and displacements all have both magnitude and direction. To analyze motion in two dimensions, you need to be able to combine vectors, break them apart, and find the net result. This section covers the graphical methods for doing that.

Vector Addition and Subtraction

Vector addition and subtraction techniques

There are a few graphical methods for adding vectors. Each gives the same resultant; they just set up the drawing differently.

Head-to-tail method: Place the tail of the second vector at the head of the first vector. The resultant is the vector drawn from the tail of the first to the head of the last. This works for any number of vectors chained together.

• For subtraction, reverse the direction of the vector being subtracted (flip it 180°), then add it head-to-tail as usual. The resultant still runs from the tail of the first vector to the head of the reversed vector.

Parallelogram method: Draw both vectors starting from the same point (tails together). Complete a parallelogram by sketching lines parallel to each vector. The resultant is the diagonal of the parallelogram, drawn from the shared tail to the opposite corner.

Triangle method: This is really just the head-to-tail method applied to exactly two vectors. Arrange them head-to-tail so they form two sides of a triangle. The third side, from the tail of the first to the head of the second, is the resultant.

All of these methods are performed within a coordinate system (usually x-y axes), which gives you a reference frame for measuring angles and directions.

Resultant vector calculations

Once you've drawn the resultant graphically, you need its magnitude and direction.

Graphical measurement:

• Measure the length of the resultant with a ruler and convert using your scale (e.g., 5 cm = 10 N).
• Measure the angle between the resultant and a reference axis (typically the positive x-axis) with a protractor.

Analytical calculation from components:

If you know the x- and y-components of the resultant, you can calculate both magnitude and direction without measuring.

1. Find the magnitude using the Pythagorean theorem: $R = \sqrt{R_x^2 + R_y^2}$ For example, if $R_x = 3$ N and $R_y = 4$ N: $R = \sqrt{3^2 + 4^2} = \sqrt{25} = 5 \text{ N}$

2. Find the direction using the arctangent function: $\theta = \tan^{-1}\left(\frac{R_y}{R_x}\right)$ Using the same components: $\theta = \tan^{-1}\left(\frac{4}{3}\right) \approx 53°$ This angle is measured from the positive x-axis.

Watch out: the arctangent function only gives angles in the first and fourth quadrants. If your vector points into the second or third quadrant (negative $R_x$), you'll need to add 180° to get the correct angle.

Vector component resolution

Any vector can be split into perpendicular components along the x- and y-axes. This is one of the most useful techniques in all of introductory physics.

Breaking a vector into components:

Given a vector with magnitude $A$ at angle $\theta$ above the positive x-axis:

1. Horizontal component: $A_x = A \cos\theta$
2. Vertical component: $A_y = A \sin\theta$

For example, a 10 N force at 30° above horizontal:

• $A_x = 10 \cos 30° \approx 8.7$ N
• $A_y = 10 \sin 30° = 5.0$ N

These formulas assume the angle is measured from the positive x-axis. If the angle is given from a different reference direction (like "north of east" or "below horizontal"), sketch it first and adjust your trig accordingly.

Reconstructing a vector from components:

1. Find the magnitude: $A = \sqrt{A_x^2 + A_y^2}$ $A = \sqrt{8.7^2 + 5.0^2} \approx 10 \text{ N}$

2. Find the angle: $\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$ $\theta = \tan^{-1}\left(\frac{5.0}{8.7}\right) \approx 30°$

This round-trip (vector → components → vector) is a good way to check your work.

Vector and Scalar Quantities

• Vectors have both magnitude and direction (e.g., velocity, force, displacement).
• Scalars have magnitude only (e.g., speed, mass, temperature).
• Unit vectors have a magnitude of exactly 1 and point in a specific direction. They're used to indicate direction without affecting magnitude.
• Equilibrium occurs when the vector sum of all forces on an object equals zero, meaning there is no net force and no acceleration.