Vectors: magnitude and direction

Scalar vs vector

A scalar is a pure amount: 5 kg, 20°C, $12. A vector carries amount and direction: 5 km east, a wind blowing 20 km/h from the north, a push diagonally up a ramp. If you only keep the size and throw away the direction, you lose the ability to aim, navigate, or balance forces.

Where vectors show up outside class

• Walking directions — “Three blocks north, then two east” are components; the straight-line shortcut is the magnitude of the resultant.
• Phone GPS / maps — displacement from where you are to a pin is a vector; distance is its length.
• Sports — a pass has speed and direction; only speed is not enough to describe the play.
• Wind and current — pilots and captains combine velocity vectors to see the true path over ground.
• Game movement — character velocity is often stored as (vx, vy) each frame.
• Forces — weight pulls down; a cable pulls along its line; engineers add force vectors to see the net effect.

Components and magnitude

In the plane we write ⟨x, y⟩: the eastward part and the northward part. The magnitude (length of the arrow) is

|v| = √(x² + y²)

That is the Pythagorean theorem—the same geometry as a ladder leaning on a wall. The classic 3-4-5 triangle is a vector magnitude in disguise: ⟨3, 4⟩ has length 5.

Adding vectors

Add components separately: ⟨a, b⟩ + ⟨c, d⟩ = ⟨a+c, b+d⟩. Picture placing the tail of the second arrow at the tip of the first; the sum runs from the start of the first to the end of the second. That is how combined displacements or combined forces are built.

Scaling

Multiply by a number k: k⟨x, y⟩ = ⟨kx, ky⟩. Positive k longer or shorter in the same direction; negative k reverses direction. Magnitude becomes |k| times the original length—length itself is never negative.

Related

• Polar coordinates (direction + distance packaging)
• Trigonometry basics