General Physics 1.4 — Vectors and Scalars

Vectors vs Scalars

A scalar is a quantity that has only magnitude — a single number with units. Examples: temperature (25°C), speed (60 km/h), mass (5 kg), energy (100 J).

A vector is a quantity that has both magnitude and direction. Examples: velocity (60 km/h east), displacement (5 m upward), force (10 N at 45° above horizontal), acceleration.

The key: if you care about which way, it’s a vector. If magnitude alone tells the full story, it’s a scalar.

Trigonometry Foundations — SOH CAH TOA

For a right triangle with angle θ:

SOH: sin(θ) = opposite / hypotenuse
CAH: cos(θ) = adjacent / hypotenuse
TOA: tan(θ) = opposite / adjacent

These ratios let you recover unknown sides and angles. If you know the hypotenuse is 10 m and the angle is 30°:

• opposite = 10 × sin(30°) = 10 × 0.5 = 5 m
• adjacent = 10 × cos(30°) = 10 × 0.866 = 8.66 m

The Pythagorean Theorem

For a right triangle with legs a and b, and hypotenuse c:

c² = a² + b²
c = √(a² + b²)

Example: legs are 3 m and 4 m.

• c = √(9 + 16) = √25 = 5 m

This is the foundation for vector magnitude: if a vector has components (x, y), its magnitude is √(x² + y²).

When You Don’t Know Opposite and Adjacent

If you have a right triangle but only know:

• The hypotenuse and one angle → use sine or cosine to find the missing side
• Two sides (but not hypotenuse) → use Pythagorean theorem to find the third
• One side and one angle → use tan or sin/cos as needed

Example: hypotenuse = 10 m, angle = 25°. Find opposite.

• opposite = 10 × sin(25°) = 10 × 0.423 = 4.23 m
• adjacent = 10 × cos(25°) = 10 × 0.906 = 9.06 m
• Check: √(4.23² + 9.06²) = √(17.9 + 82.1) = √100 = 10 ✓

The trigonometric functions bridge the gap when direct measurement isn’t available.

Vector Components

A vector can be decomposed into perpendicular components (usually x and y in 2D):

Vector magnitude: V = √(Vx² + Vy²)
Component Vx = V × cos(θ)
Component Vy = V × sin(θ)

Where θ is the angle from the horizontal (x-axis).

Example: A force vector of 50 N at 35° above horizontal.

• Fx = 50 × cos(35°) = 50 × 0.819 = 40.95 N
• Fy = 50 × sin(35°) = 50 × 0.574 = 28.7 N
• Check magnitude: √(40.95² + 28.7²) = √(1677 + 823.7) = √2500.7 ≈ 50 N ✓

Vector Addition and Subtraction

Component method (most practical):

1. Break each vector into x and y components
2. Add components: Rx = Ax + Bx, Ry = Ay + By
3. Recombine: R = √(Rx² + Ry²), angle = arctan(Ry / Rx)

Example: Add vectors A (30 N at 0°) and B (20 N at 90°).

• Ax = 30, Ay = 0
• Bx = 0, By = 20
• Rx = 30, Ry = 20
• Resultant = √(900 + 400) = √1300 = 36.06 N at arctan(20/30) = 33.7° ✓

Subtraction works the same way, but subtract components: Rx = Ax − Bx.

Dot Product (Scalar Product)

Multiplying two vectors gives a scalar:

A · B = |A| |B| cos(θ)

Where θ is the angle between them. Alternatively, using components:

A · B = (Ax × Bx) + (Ay × By)

Intuition: dot product measures “how much” two vectors point in the same direction. If perpendicular, dot product is zero.

What Stuck

The insight that trigonometry is fundamentally about recovering unknown quantities from known ones in a triangle. SOH CAH TOA isn’t arbitrary — it’s the inevitable bridge between angles and side lengths. Vector components are just applying this same principle to arbitrary directions, not just right triangles. Once you have components, every operation becomes algebra.