Skill: vectors

Vectors

Master arrows with size and direction: components, magnitude, angles, addition, dot products, and real navigation and force problems.

1 Components and magnitude

A vector is an arrow: it has a length (magnitude) and a direction. Component form ⟨x, y⟩ splits it into legs.

The triangle relations

|v| = √(x² + y²)   (magnitude is a hypotenuse)
θ = atan2(y, x)   (direction angle, 0° ≤ θ < 360°)
x = |v| cos θ, y = |v| sin θ   (rebuild components)

The vector from A = (a₁, a₂) to B = (b₁, b₂) is ⟨b₁ − a₁, b₂ − a₂⟩ — “head minus tail.”

Example Magnitude of ⟨6, 8⟩

|v| = √(36 + 64) = √100 = 10.

2 Adding, scaling, and dotting

Vector arithmetic happens component by component — except the dot product, which collapses to a single number.

The operations

u + v = ⟨u₁ + v₁, u₂ + v₂⟩   (tip-to-tail)
c·v = ⟨c v₁, c v₂⟩   (scales length; c < 0 flips direction)
u · v = u₁v₁ + u₂v₂   (a number, not a vector)

The dot product also equals |u||v| cos φ, where φ is the angle between them — so u · v = 0 means perpendicular.

Example Combine u = ⟨2, −1⟩, v = ⟨3, 4⟩

u + v = ⟨5, 3⟩; 2u = ⟨4, −2⟩; u · v = 2·3 + (−1)·4 = 2.

3 Navigation and force resultants

Bearings and forces are where vectors earn their keep: add the arrows, read the result.

Bearings

A bearing is measured clockwise from north. N30°E means face north, turn 30° toward east — in component form that’s ⟨|v| sin 30°, |v| cos 30°⟩ (east is x, north is y). Add the vectors tip-to-tail; the resultant is the single vector that does the same job.

Example Resultant of ⟨3, 4⟩ and ⟨5, −1⟩

Add components: ⟨3 + 5, 4 + (−1)⟩ = ⟨8, 3⟩. Magnitude √(64 + 9) = √73 ≈ 8.54; direction atan2(3, 8) ≈ 20.6°.

4 Worked examples

Core computations, each verified independently.

Example 1 Magnitude

Find |⟨3, 4⟩|.

√(9 + 16) = √25 = 5.

Example 2 Direction angle

Find the direction angle of ⟨1, 1⟩ (degrees).

atan2(1, 1) = 45°.

Example 3 Addition

Compute ⟨1, 2⟩ + ⟨3, 4⟩.

⟨1 + 3, 2 + 4⟩ = ⟨4, 6⟩.

Example 4 Dot product

Compute ⟨2, 3⟩ · ⟨4, −1⟩.

2·4 + 3·(−1) = 8 − 3 = 5.

Example 5 Scalar multiple

Compute 2⟨3, 4⟩.

⟨2·3, 2·4⟩ = ⟨6, 8⟩.

5 Common mistakes

Three errors that break vector work.

Adding magnitudes instead of components

Wrong: |⟨3, 4⟩| + |⟨5, 12⟩| = 5 + 13 = 18, so the sum has magnitude 18.
Right: add components first: ⟨8, 16⟩, then |v| = √(64 + 256) = √320 ≈ 17.89. Magnitude does not distribute over addition.

Forgetting bearings start at north

Wrong: N30°E → ⟨cos 30°, sin 30°⟩.
Right: bearings rotate clockwise from north: N30°E = ⟨sin 30°, cos 30°⟩ (east = x, north = y).

Treating a dot product like a vector

Wrong: ⟨2, 3⟩ · ⟨4, 5⟩ = ⟨8, 15⟩.
Right: the dot product is a number: 8 + 15 = 23. u · v = 0 means the vectors are perpendicular.

6 Key vocabulary

Words to know

Component form — v = ⟨x, y⟩, the vector’s legs.
Magnitude — |v| = √(x² + y²), the vector’s length.
Direction angle — atan2(y, x), measured from the positive x-axis.
Resultant — the single vector equal to a sum of vectors.
Bearing — an angle measured clockwise from north (navigation).

7 Quick check

Try these before moving on — click to reveal each answer.

Compute 3⟨2, −1⟩ − ⟨1, 5⟩.

⟨6, −3⟩ − ⟨1, 5⟩ = ⟨5, −8⟩.

Are ⟨3, −4⟩ and ⟨4, 3⟩ perpendicular?

Dot = 12 − 12 = 0, so yes.

A boat heads N30°E at 10 mph. What is its vector?

⟨5, 8.66⟩ (east = 10 sin 30° = 5, north = 10 cos 30° ≈ 8.66).

8 Next steps

Now drill the skill with endless randomized problems.

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