A vector has both size (magnitude) and direction; a scalar has size only. In two dimensions we write a vector with the unit vectors i (1 unit in the x-direction) and j (1 unit in the y-direction): a = xi + yj.

Magnitude: |a| = √(x² + y²). Direction: the angle θ from the positive x-axis, found from tan θ = yx and the quadrant the vector points into. Going the other way, a vector of magnitude r at angle θ is r cos θ i + r sin θ j.

Worked example. a = 3i − 4j. |a| = √(9 + 16) = 5. It points right and down (fourth quadrant); the reference angle is tan⁻¹(43) = 53.1°, so θ = 360° − 53.1° = 306.9°.

Add or subtract vectors component by component, and multiply by a scalar k by multiplying both components: k(xi + yj) = kxi + kyj. Two vectors are parallel when one is a scalar multiple of the other.

The vector from point A to point B is b − a (position of B minus position of A). A unit vector has magnitude 1: divide a vector by its magnitude to get the unit vector in its direction.

Worked example. a = 2i + j and b = −i + 4j. Then a + b = i + 5j, and 3a − b = (6 + 1)i + (3 − 4)j = 7i − j.

For u = ai + bj and v = ci + dj, the scalar product (dot product) is u · v = ac + bd: multiply matching components and add. The result is a number, not a vector.

It also equals |u||v| cos θ, where θ is the angle between the vectors, so cos θ = u · v|u||v|. Two non-zero vectors are perpendicular exactly when their scalar product is 0.

Worked example. (i + 2j) · (4i − 2j) = 1 × 4 + 2 × (−2) = 0, so the two vectors are perpendicular.