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Find the Sum of
Two Vectors

Learn to combine vectors given in magnitude and direction form by breaking them into components, adding, and converting back.

Precalculus

What Is a Vector?

A vector is a quantity that has both magnitude (size) and direction (angle). Unlike a scalar, which is just a number, a vector tells you how much and which way.

Magnitude
The length of the vector, often written as \( \|\mathbf{v}\| \) or \( r \). Always non-negative.
Direction
The angle \( \theta \) measured counter-clockwise from the positive \( x \)-axis.

Visualizing a Single Vector

x y
\(\theta\)
\(r\)
\(r\cos\theta\)
\(r\sin\theta\)
 O

Why Add Vectors?

Many real-world quantities combine as vectors: forces on an object, velocities of a boat in a current, or displacements along different paths. The resultant (sum) tells you the net effect.

Physics
Net force on an object is the vector sum of all individual forces.
Navigation
A pilot's ground speed is the sum of the airspeed vector and wind vector.
Engineering
Structural loads combine as vectors to determine stress on a beam.

The Component Method

To add two vectors given in magnitude-direction form, convert each to component form, add the components, then convert back.

1
Decompose each vector into \( x \)- and \( y \)-components using cosine and sine.
2
Add the \( x \)-components together and the \( y \)-components together.
3
Find the magnitude of the resultant: \( \|\mathbf{R}\| = \sqrt{R_x^2 + R_y^2} \).
4
Find the direction: \( \theta_R = \tan^{-1}\!\left(\dfrac{R_y}{R_x}\right) \), adjusting for the correct quadrant.

Formulas at a Glance

The Parallelogram Law

\(\mathbf{u}\)
\(\mathbf{v}\)
\(\mathbf{u}+\mathbf{v}\)

Component vs. Geometric Methods

Component Method Geometric (Tip-to-Tail)
Input Magnitude & direction Magnitude & direction
Tool \( \sin, \cos, \tan^{-1} \) Ruler & protractor
Precision Exact Approximate
Best For Any number of vectors Quick visual estimate
Exam Use Primary Sketch to verify

How to Apply the Method

1
Write down each vector's magnitude \( r \) and direction \( \theta \).
2
Compute \( x = r\cos\theta \) and \( y = r\sin\theta \) for each vector. Keep extra decimal places.
3
Sum: \( R_x = \sum x_i \), \( R_y = \sum y_i \).
4
Magnitude: \( \|\mathbf{R}\| = \sqrt{R_x^2 + R_y^2} \).
5
Direction: \( \theta_R = \tan^{-1}(R_y/R_x) \). Adjust quadrant if needed.
6
Verify with a sketch: does the resultant point in the expected direction?

Example 1: A Basic Case

Find \( \mathbf{u} + \mathbf{v} \) where \( \mathbf{u} \) has magnitude 6 at \( 30^\circ \) and \( \mathbf{v} \) has magnitude 4 at \( 120^\circ \).

Example 1: Result

Result: \( \|\mathbf{R}\| \approx 7.21 \) at \( \theta \approx 63.7^\circ \)

Example 2: Different Quadrants

Find \( \mathbf{a} + \mathbf{b} \) where \( \mathbf{a} \) has magnitude 10 at \( 50^\circ \) and \( \mathbf{b} \) has magnitude 8 at \( 200^\circ \).

Example 2: Result

Result: \( \|\mathbf{R}\| \approx 5.04 \) at \( \theta \approx 102.5^\circ \)

Example 3: Perpendicular Vectors

Find \( \mathbf{p} + \mathbf{q} \) where \( \mathbf{p} \) has magnitude 5 at \( 0^\circ \) and \( \mathbf{q} \) has magnitude 12 at \( 90^\circ \).

\(\mathbf{p}\)
\(\mathbf{q}\)
\(\mathbf{R}\)

Common Mistakes

✕ Adding Magnitudes Directly
You cannot simply add \( r_1 + r_2 \) unless the vectors point in the same direction. Use components instead.
✕ Wrong Quadrant
\( \tan^{-1} \) returns values in \( (-90^\circ, 90^\circ) \). You must add \( 180^\circ \) when \( R_x < 0 \).
✕ Degrees vs. Radians
Make sure your calculator is in the correct mode. A \( \cos(30) \) in radian mode gives a completely different answer.

Summary of Key Steps

Step Action Formula
1 Decompose \( x = r\cos\theta,\; y = r\sin\theta \)
2 Add \( R_x = \sum x_i,\; R_y = \sum y_i \)
3 Magnitude \( \|\mathbf{R}\| = \sqrt{R_x^2 + R_y^2} \)
4 Direction \( \theta_R = \tan^{-1}(R_y / R_x) \)
5 Quadrant Fix Add \( 180^\circ \) if \( R_x < 0 \)

Question 1 of 5

Vector \( \mathbf{u} \) has magnitude 8 at \( 0^\circ \) and vector \( \mathbf{v} \) has magnitude 6 at \( 90^\circ \). What is \( \|\mathbf{u}+\mathbf{v}\| \)?

Question 2 of 5

What is the \( x \)-component of a vector with magnitude 10 at \( 60^\circ \)?

Question 3 of 5

After adding components you get \( R_x = -3 \) and \( R_y = 4 \). In which quadrant does the resultant lie?

Question 4 of 5

Two vectors both have magnitude 5. One points at \( 0^\circ \) and the other at \( 180^\circ \). What is the magnitude of their sum?

Question 5 of 5

Given resultant components \( R_x = -3 \) and \( R_y = 4 \), a calculator returns \( \tan^{-1}(R_y/R_x) \approx -53.1^\circ \). What is the correct angle for the resultant vector?