When several vectors act at once, their resultant is their sum. Add components, never magnitudes.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Why magnitudes fail
A force of 3 N east and 4 N north give a resultant of 5 N, not 7 N. Direction is part of the arithmetic.
Resolving
A vector of magnitude v at angle θ has components (v cos θ, v sin θ).
Going back
From components, the magnitude is √(x² + y²) and the direction is arctan(y ÷ x), adjusted for the quadrant.
Equilibrium
An object is in equilibrium when the forces sum to the zero vector, meaning both component totals are zero.
Navigation
A plane’s ground velocity is its air velocity plus the wind velocity, which is why a crosswind changes both speed and heading.
The resultant is a vector sum
Two forces acting on an object combine into one resultant, found by adding the vectors. Breaking each force into components, adding componentwise, and recombining is the reliable procedure.
Resolve into components first
A force of 50 N at 30° becomes ⟨50cos30°, 50sin30°⟩. Component form makes addition trivial; adding magnitudes and angles directly is simply wrong, and it is the error these problems are designed to catch.
Headings and drift
An aircraft's ground velocity is its air velocity plus the wind. Solving for the heading needed to reach a destination is a vector equation, and it is why navigation is done with vectors rather than arithmetic.
Equilibrium means the sum is zero
An object at rest has forces summing to the zero vector, so the components sum to zero in each direction separately. That gives two equations from one condition, which is usually enough to solve for unknown forces.
Step 2: Try It Yourself
Tap and try it out.
- Vector a(3, 0)
- Vector b(0, 4)
- a + b(3, 4)
The dashed arrow is b again, moved to the tip of a. The sum closes the triangle, and its components are just the x parts added and the y parts added.
Step 3: Watch an Example
One step at a time.
Watch Tomas Find a Ground Speed
A plane flies east at 200 km/h through a 50 km/h northward wind.
- Step 1
The air velocity is (200, 0) and the wind is (0, 50).
Step 4: Your Turn
Practice makes it stick.
The Resultant
Problem 1 of 2
Forces (3, 0) N and (0, 4) N act together. What is the magnitude of the resultant, in newtons?
N
The Balance
Problem 2 of 2
Two forces are (6, −2) N and (−6, 2) N. What is the magnitude of the resultant, in newtons?
N
Add the Arrows
1 of 8
(5, 2) + (1, 6). What is the y component?
2 of 8
Magnitude of (6, 8)?
3 of 8
A vector of magnitude 10 at 0°. What is its x component?
4 of 8
A vector of magnitude 10 at 90°. What is its x component?
5 of 8
A boat rows east at 12 and a current pushes north at 5. Resultant speed?
6 of 8
Forces (4, 3) and (−4, −3). Magnitude of the resultant?
7 of 8
Which quantities are vectors?
8 of 8
A plane flies north at 300 with a 40 westward wind. Resultant speed, to the nearest whole number?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Forces (0, 5) and (12, 0) act together. Magnitude of the resultant?
Question 2 of 2
Why can magnitudes not simply be added?
What You Learned
- Add vectors by components; magnitudes do not add.
- A vector of magnitude v at angle θ has components (v cos θ, v sin θ).
- Equilibrium means every component total is zero.