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The Language of Vectors — and Where It Takes You

Direction matters. Learn to split any vector into components, add vectors two ways, and use the unit-vector notation that kinematics, dynamics, and fields are all written in — the doorway to the rest of physics.

Scalars and vectors: when direction matters

Some quantities are fully described by a single number with a unit: a mass of 3 kg, a time of 5 s, a temperature of 300 K. These are scalars — magnitude only, nothing more to say.

Others make no sense without a direction. Saying you walked 3 km is incomplete; 3 km north takes you somewhere quite different from 3 km east. Quantities with both a magnitude and a direction are vectors — displacement, velocity, acceleration, force. We draw a vector as an arrow — its length is the magnitude, the way it points is the direction — and write it \vec{A}, with magnitude |\vec{A}| or simply A.

Resolving a vector into components

Here is the workhorse trick of all physics: replace one awkward arrow with two convenient ones lying along the axes of a coordinate system. A vector \vec{A} pointing at angle \theta above the x-axis has an x-component and a y-component — its vector components.

A vector resolved into perpendicular x- and y-components, with its magnitude and angle.

A_x = A\cos\theta, \qquad A_y = A\sin\theta

Project the arrow onto each axis with cosine and sine.

The two components carry all the information, so you can always rebuild the arrow from them: the Pythagorean theorem gives back the magnitude, and the arctangent gives back the direction.

A = \sqrt{A_x^{2} + A_y^{2}}, \qquad \theta = \tan^{-1}\!\left(\frac{A_y}{A_x}\right)

From components back to magnitude and angle.

Adding vectors — arrows and components

Geometrically, add vectors tip to tail: draw the second arrow starting where the first one ends, and the resultant runs from the very beginning to the very end (equivalently, it is the diagonal of the parallelogram the two vectors span).

Interactive: drag two vectors and watch the resultant form by the tip-to-tail rule, with live components and magnitude-and-angle.

For actual calculation, the component method is exact and easy: add the x-components together, add the y-components together, then rebuild the resultant from those sums.

\vec{R} = \vec{A} + \vec{B}: \quad R_x = A_x + B_x, \quad R_y = A_y + B_y

Add like components; never add the magnitudes directly.

It is tidy to name one-length arrows along the axes: \hat{\imath} points along +x and \hat{\jmath} along +y. These are unit vectors, and with them any vector is just \vec{A} = A_x\hat{\imath} + A_y\hat{\jmath} — so adding vectors becomes nothing more than adding like terms.

\vec{A} = A_x\hat{\imath} + A_y\hat{\jmath}

Unit-vector notation: a vector as a sum of components along the axes.

  1. Worked example. A hiker walks 4.0 km east, then 3.0 km at 30^\circ north of east. Find the total displacement. First choose axes: let x point east and y point north.
  2. Leg 1 (4.0 km east) has components (4.0,\ 0)\ \mathrm{km}.
  3. Leg 2 (3.0 km at 30^\circ): A_x = 3.0\cos30^\circ = 2.60 km, A_y = 3.0\sin30^\circ = 1.50 km.
  4. Add components: R_x = 4.0 + 2.60 = 6.60 km, R_y = 0 + 1.50 = 1.50 km.
  5. Magnitude: R = \sqrt{6.60^{2} + 1.50^{2}} \approx 6.8 km. Direction: \theta = \tan^{-1}(1.50/6.60) \approx 13^\circ north of east.
  6. Sanity check: the answer 6.8 km is less than the 7.0 km you would wrongly get by adding the lengths — exactly because the two legs do not point the same way.

Where vectors take you next

You now hold the grammar of physics. From here the ladder is built almost entirely on vectors: kinematics describes motion with the vectors displacement, velocity, and acceleration; dynamics adds force and Newton's laws; and later tracks turn the electric and magnetic fields into vectors defined at every point in space.

Everything from this whole track rides along: quantities carry units, answers are checked by dimensions and bracketed by estimates, and every measurement comes with an honest uncertainty. With that toolkit in hand, you are ready to start describing motion.