Vectors and Pythagoras

A scalar is a number only: 120 V magnitude with no direction, 10 A, 50 ft of wire. A vector adds direction: 120 V leading current by 30°, a force northeast at 50 lb, a displacement 3 ft east plus 4 ft north.

In AC, voltage and current are often vectors because they can point different ways on a time diagram (phase). Even before phase, any two effects at right angles combine with Pythagoras—not by straight addition.

In any right triangle:

a² + b² = c²

where c is the hypotenuse (the longest side, opposite the 90° corner) and a and b are the two shorter sides.

Example. a = 3, b = 4 c² = 3² + 4² = 9 + 16 = 25 c = √25 = 5

The classic 3-4-5 triangle appears constantly in layout and in vector sketches. Scale it: 6-8-10, 9-12-15—same shape.

To add two vectors that are 90° apart:

1. Draw one vector along the x-axis (horizontal).
2. Draw the second from the tip of the first, vertical.
3. The resultant runs from the start of the first to the end of the second—the hypotenuse.

Magnitude: R = √(A² + B²) Direction: θ = tan⁻¹(B / A) from the horizontal reference.

Example. Horizontal component 6 V, vertical component 8 V. R = √(36 + 64) = √100 = 10 V θ = tan⁻¹(8/6) ≈ 53.1°

Head-to-tail method still works: slide the second vector so its tail touches the first vector's head. The resultant is the closing line. If the angle is not 90°, use trigonometry (law of cosines) or a scale drawing. In this course, most field examples start with right-angle combinations—the Pythagorean case.

Subtracting vector B from A is the same as adding −B (same length, opposite direction). In AC troubleshooting you sometimes compare before and after a change: the difference between two phasors tells you what a capacitor or inductor added to the circuit.

Imagine two voltage drops in a branch: one purely resistive (in phase with current) and one purely reactive (90° shifted)—you will study those parts later. Their total drop is not 6 V + 8 V = 14 V if they are orthogonal; it is √(6² + 8²) = 10 V. Treating AC drops like DC addition is one of the most expensive mistakes in the trade.

Vectors carry the same units as the quantity: volts, amperes, ohms (as impedance magnitude), or feet for mechanical analogies. When adding, both components must use consistent units and the same reference direction. A sketch at 1 cm = 2 V is fine on paper; on site, use the same ratio on every leg.

Any vector can be split into two perpendicular components—often horizontal (x) and vertical (y). If magnitude is M and angle from horizontal is θ:

• Horizontal component: M × cos θ
• Vertical component: M × sin θ

This is the reverse of adding components. A 10 V vector at 53.1° has horizontal 6 V and vertical 8 V—matching our earlier example. In AC, you will resolve impedance and voltage into real and reactive parts, then recombine with Pythagoras for totals.

Push a cart 3 ft east, then 4 ft north. You did not travel 7 ft from start—you are 5 ft away in a straight line northeast. Walk the same path with a tape: the diagonal is the resultant displacement. Electrical vectors obey the same geometry; only the units change from feet to volts or amperes.

• Resultant must be longer than either perpendicular leg alone (unless one leg is zero).
• For equal legs at 90°, resultant = leg × √2 (45° case).
• If R seems smaller than one component, you probably subtracted when you should have combined orthogonal parts with Pythagoras.
• Dimensions must match: never add volts directly to amperes—vectors combine only when they represent the same kind of quantity in the same diagram.

Situation. A motor branch shows 6 V drop measured in phase with current on the feeder and 8 V drop on a reactive part of the same path (measured 90° shifted on a scope demo during training). A helper adds 6 + 8 and reports 14 V total drop—then wonders why the math disagrees with the phasor diagram.

Correct approach. Perpendicular components: Total = √(6² + 8²) = 10 V

Applied lesson. Orthogonal components combine with Pythagoras. Parallel readings on a multimeter alone cannot always reveal the 90° split—you need to know what kind of drop each measurement represents. This course builds that distinction lesson by lesson.