Scalars and vectors
Some quantities need only a number and a unit: a mass of 2 kg, a temperature of 20 °C, a time of 5 s. These are scalars. Others need a direction as well: a force of 10 N pointing east, a velocity of 3 m/s upward. These are vectors, described by a magnitude (size) and a direction.
The distinction matters for addition. Scalars add as ordinary numbers. Vectors add with direction taken into account, so two 10 N forces can total 20 N, 0 N, or something in between, depending on how they point.
Two ways to describe a vector
A vector can be given as magnitude and angle, or as components along chosen axes, typically horizontal (x) and vertical (y). The components of a vector with magnitude F at angle θ above the x-axis are:
- Fₓ = F cos θ
- F_y = F sin θ
Going back, magnitude = √(Fₓ² + F_y²) and the angle satisfies tan θ = F_y / Fₓ. These rely on right-angle trigonometry.
Example: a 20 N force at 30° above the x-axis has components Fₓ = 20 × 0.866 ≈ 17.3 N and F_y = 20 × 0.5 = 10 N. Check: 17.32² + 10² ≈ 300 + 100 = 400, and the square root is 20. It agrees.
The core method: add components
For vector addition in components, add all the x components together and all the y components together. Then rebuild the result if you need magnitude and direction.
Example 1: a right angle
A 3 N force acts along x and a 4 N force acts along y. The components of the total are (3, 4).
- Magnitude = √(3² + 4²) = √25 = 5 N
- Angle = arctan(4/3) ≈ 53° above the x-axis
Simply adding the sizes would give 7 N, which is wrong because the directions differ.
Example 2: components given
Force A = (6, 2) N and force B = (−2, 5) N, where a negative component points the other way.
- Total x = 6 + (−2) = 4
- Total y = 2 + 5 = 7
- Magnitude = √(4² + 7²) = √(16 + 49) = √65 ≈ 8.06 N
- Angle = arctan(7/4) = arctan(1.75) ≈ 60.3° above the x-axis
Quick check: tan 60° ≈ 1.732, and tan 61° is a little over 1.80, so 1.75 lies between them, which is consistent with 60.3°.
Example 3: one force given as magnitude and angle
Force C is 10 N along the x-axis, so C = (10, 0). Force D is 10 N at 90°, so D = (0, 10). The total is (10, 10), with magnitude √200 ≈ 14.1 N at 45°. Notice 14.1 is much less than 20, the sum of the magnitudes.
Picturing it: tip to tail
Graphically, draw the first vector, then start the second at its tip. The total runs from the start of the first to the tip of the last. This works for any number of vectors, and the order does not change the result. The picture helps you sanity check: the answer should lie between the extremes of fully aligned (sizes add) and fully opposed (sizes subtract).
Velocities and directions
The same method works for any vector quantity. A boat steers at 4 m/s straight across a river while the current carries it 3 m/s downstream. The components relative to the bank are 4 across and 3 along the bank.
- Speed over the ground = √(4² + 3²) = 5 m/s
- It drifts at an angle with tan = 3/4, so about 37° from straight across.
For a displacement, adding successive legs of a journey follows the same rules. Walking 3 m east then 4 m north leaves you 5 m from where you started, though you walked 7 m.
Equilibrium
When an object is not accelerating, the vector sum of the forces on it is zero. Test: forces (3, 4) N, (−3, 0) N and (0, −4) N sum to x: 3 − 3 + 0 = 0 and y: 4 + 0 − 4 = 0, so they balance.
This is how tension in cables, loads on supports and forces on joints are first analysed: sum the components in each direction and set them to zero. Real structures need allowances, load factors and review by a qualified engineer, and the rules differ across countries and industries, so treat these examples as concept practice only.
Subtraction and scaling
Subtracting a vector means adding its opposite, which has every component reversed. Doubling a vector doubles each component. For instance, (6, 2) − (−2, 5) = (6 + 2, 2 − 5) = (8, −3).
Common slips
- Adding magnitudes when directions differ.
- Sign errors in components, particularly angles measured from a different axis or in the second, third or fourth quadrant. Draw a quick sketch.
- Calculator mode. Use degrees or radians consistently.
- Arctan quadrant errors. A calculator’s arctan cannot tell (3, 4) from (−3, −4), so check which way the vector actually points.
- Mixed units. Convert first, ideally using good unit conversion habits, and confirm the result with dimensional analysis.
Vectors extend to three dimensions by adding a z component with the same rules. Two further operations, the dot product and the cross product, handle angles between vectors, work and moments, and can be learned once adding in components feels natural. Keep sensible significant figures in your answers, and treat these examples as illustrations with chosen numbers.