chevron_left Vector Addition chevron_right

Vector Addition: Finding the Combined Effect

What Are Vectors and Scalars?

Before we can add vectors, we must understand what they are. In science and math, we deal with two types of quantities: scalars and vectors.

A scalar is a quantity that has only magnitude (size or amount). Examples include temperature, mass, time, and speed. For instance, saying "the car's speed is 60 km/h" is a scalar quantity; it tells us how fast but not in which direction.

A vector is a quantity that has both magnitude and direction. Examples include displacement, velocity, force, and acceleration. Saying "the car's velocity is 60 km/h north" is a vector quantity; it tells us both how fast and where it's going. Vectors are often represented by arrows. The length of the arrow represents the magnitude, and the direction the arrow points represents the direction of the quantity.

Graphical Methods for Vector Addition

Since vectors have direction, we cannot simply add their magnitudes like we do with scalars. Adding a 3 km walk east and a 4 km walk north does not equal a 7 km walk! The two main graphical methods for adding vectors are the tip-to-tail and parallelogram methods.

The Tip-to-Tail Method

This is the most common and intuitive method. To add two vectors, A and B:

1. Draw the first vector, A, to scale (e.g., 1 cm = 1 km).
2. Place the tail of the second vector, B, at the tip (head) of the first vector, A.
3. The resultant vector, R, is drawn from the tail of the first vector to the tip of the last vector.

Example: A person walks 5 m east (A) and then 3 m north (B). The resultant displacement (R) is the straight-line path from the starting point to the ending point. You can measure the length and direction of R on your drawing to find the magnitude and direction of the resultant.

The Parallelogram Method

This method is useful when vectors are acting from a common point. To add vectors A and B:

1. Draw both vectors from the same origin point.
2. Complete the parallelogram by drawing lines parallel to each vector.
3. The resultant vector R is the diagonal of the parallelogram drawn from the common origin.

Both graphical methods will yield the same resultant vector. The tip-to-tail method is generally preferred for adding more than two vectors.

The Analytical Method: Adding Vectors with Components

While graphical methods are helpful for visualization, the analytical method provides precise numerical answers. This method involves breaking down vectors into their components^[1].

Any vector in a two-dimensional plane can be resolved into an x-component (horizontal) and a y-component (vertical). If a vector V has a magnitude $V$ and makes an angle $\theta$ with the positive x-axis, its components are:

• x-component: $V_x = V \cos \theta$
• y-component: $V_y = V \sin \theta$

Example: Add vector A (10 units, 30°) and vector B (8 units, 120°).

Step 1: Resolve into components.

• $A_x = 10 \cos 30^\circ = 10 \times 0.866 = 8.66$
• $A_y = 10 \sin 30^\circ = 10 \times 0.5 = 5.00$
• $B_x = 8 \cos 120^\circ = 8 \times (-0.5) = -4.00$
• $B_y = 8 \sin 120^\circ = 8 \times 0.866 = 6.93$

Step 2 & 3: Sum the components.

• $R_x = A_x + B_x = 8.66 + (-4.00) = 4.66$
• $R_y = A_y + B_y = 5.00 + 6.93 = 11.93$

Step 4 & 5: Find magnitude and direction.

• $R = \sqrt{(4.66)^2 + (11.93)^2} = \sqrt{21.72 + 142.32} = \sqrt{164.04} \approx 12.81$ units
• $\theta = \tan^{-1}(11.93 / 4.66) = \tan^{-1}(2.56) \approx 68.7^\circ$ (measured from the positive x-axis)

Practical Applications of Vector Addition

Vector addition is not just a mathematical exercise; it is used constantly in science and engineering to predict outcomes and design systems.

Navigation: An airplane pilot must account for the plane's velocity and the wind's velocity. The plane's path relative to the ground is the vector sum of its airspeed and the wind speed. If a plane flies due north at 200 km/h but encounters a crosswind blowing east at 50 km/h, the resultant ground velocity will be northeast. The pilot must continuously adjust the heading to compensate for the wind and stay on the desired course.

Forces in Engineering: When building a bridge, engineers must calculate the net force acting on various parts. If two cables are holding up a light post, each pulling with a different force and at a different angle, the resultant force on the post is the vector sum of the two cable tensions. This ensures the structure is stable and can handle the combined loads.

Video Game Physics: In video games, the motion of characters and objects is governed by vectors. When a character is hit by an explosion, the game engine calculates a "force vector" for the blast. If the character is already moving, their new velocity vector is the sum of their current velocity vector and the blast force vector, resulting in a realistic-looking change in motion.

Common Mistakes and Important Questions

Footnote

^[1] Components: The projections of a vector along the axes of a coordinate system. A two-dimensional vector V can be expressed as the sum of its x-component ($V_x$) and y-component ($V_y$), often written as V = ($V_x$, $V_y$).