3.3 Vector Addition and Subtraction: Analytical Methods

Vector Addition and Subtraction

Vector addition and subtraction let you combine or compare quantities that have both magnitude and direction, like forces, velocities, or displacements. The analytical method (also called the component method) is more precise than drawing diagrams, and it's the approach you'll rely on for most problem-solving in this course.

The core idea: break each vector into x- and y-components, work with those components separately, then recombine them into a single resultant vector.

Principles of Vector Addition

Vectors are represented by arrows. The arrow's length shows the magnitude, and the arrowhead points in the direction. Unlike scalars (which are just numbers), you can't add vectors by simply adding their magnitudes. A 3 N force pointing east plus a 4 N force pointing north does not equal 7 N. You need to account for direction.

The component method handles this by splitting each vector into horizontal (x) and vertical (y) parts, then working with each direction independently.

To add vectors analytically:

1. Resolve each vector into its x- and y-components.

2. Add all the x-components together to get $R_x$.

3. Add all the y-components together to get $R_y$.

4. Use $R_x$ and $R_y$ to find the magnitude and direction of the resultant vector.

To subtract vectors analytically (e.g., $\vec{A} - \vec{B}$):

Subtraction works the same way, except you reverse the second vector. That means you subtract its components instead of adding them:

1. $R_x = A_x - B_x$

2. $R_y = A_y - B_y$

3. Find the magnitude and direction from $R_x$ and $R_y$ just as you would for addition.

This is equivalent to adding $\vec{A} + (-\vec{B})$, where $-\vec{B}$ is a vector with the same magnitude as $\vec{B}$ but pointing in the opposite direction.

Analytical Vector Components

To use the component method, you first need to decompose each vector. This process is called orthogonal decomposition, meaning you break a vector into two perpendicular parts along the x- and y-axes.

For a vector with magnitude $A$ at angle $\theta$ measured counterclockwise from the positive x-axis:

• x-component: $A_x = A \cos \theta$
• y-component: $A_y = A \sin \theta$

For example, a velocity of 50 m/s at 30° above the positive x-axis has:

• $v_x = 50 \cos 30° = 43.3 \text{ m/s}$
• $v_y = 50 \sin 30° = 25.0 \text{ m/s}$

Signs of components depend on the quadrant:

• Quadrant I (up and to the right): $A_x > 0$, $A_y > 0$
• Quadrant II (up and to the left): $A_x < 0$, $A_y > 0$
• Quadrant III (down and to the left): $A_x < 0$, $A_y < 0$
• Quadrant IV (down and to the right): $A_x > 0$, $A_y < 0$

You don't need to memorize these sign rules separately. If you consistently measure $\theta$ counterclockwise from the positive x-axis, the cosine and sine functions will automatically give you the correct signs.

Resultant Vector Characteristics

Once you have the resultant components $R_x$ and $R_y$, you need to convert them back into a magnitude and direction.

Magnitude (using the Pythagorean theorem):

$R = \sqrt{R_x^2 + R_y^2}$

Direction (using inverse tangent):

$\theta = \tan^{-1}\left(\frac{R_y}{R_x}\right)$

Watch out for quadrant ambiguity. Your calculator's $\tan^{-1}$ function only returns angles between $-90°$ and $+90°$, which covers Quadrants I and IV. If the resultant vector is in Quadrant II or III (meaning $R_x < 0$), you need to add $180°$ to the calculator's output to get the correct angle.

Vector and Scalar Quantities

• A vector has both magnitude and direction (displacement, velocity, acceleration, force).
• A scalar has only magnitude (temperature, mass, speed, time).

Unit vectors are vectors with a magnitude of exactly 1, used purely to indicate direction. In the standard coordinate system, $\hat{i}$ points along the positive x-axis and $\hat{j}$ points along the positive y-axis. Any 2D vector can be written in unit vector notation as $\vec{A} = A_x\hat{i} + A_y\hat{j}$, which is just another way of expressing its components.