Vectors in 2D and 3D

Scalars and vectors

Some physical quantities need only a number and a unit. These are scalars. Mass, temperature, time, energy, and speed are common scalar quantities. Saying an object has a mass of $5 kg$ is complete.

Other quantities require direction as well as size. These are vectors. Displacement, velocity, acceleration, force, momentum, and electric field are vector quantities. Saying a car has a velocity of $20 m/s$ is incomplete unless you include direction, such as east or north.

A vector has magnitude and direction. Magnitude means size or length. Direction tells where it points.

Vector notation

Vectors are often written with arrows, bold letters, or component notation. For example, a force vector may be written as ec{F} or bold $F$. In text-based code or typed notes, arrows are often represented symbolically.

In two dimensions, a vector can be written in component form:

angle$$ In three dimensions: $$ec{A} = langle A_x, A_y, A_z angle$$ The components describe how much of the vector points along each coordinate axis. ## Unit vectors Unit vectors have magnitude 1 and point along coordinate directions. The standard unit vectors are often written as $hat{i}$, $hat{j}$, and $hat{k}$ for the x, y, and z directions. A three-dimensional vector can be written as $$ec{A} = A_xhat{i} + A_yhat{j} + A_zhat{k}$$ This notation is useful because it separates direction from amount. ## Magnitude in two dimensions The magnitude of a 2D vector follows from the Pythagorean theorem: $$|ec{A}| = sqrt{A_x^2 + A_y^2}$$ If a displacement has components $3 m$ east and $4 m$ north, its magnitude is $5 m$. Direction can be found with trigonometry: $$ heta = an^{-1}left(rac{A_y}{A_x} ight)$$ with attention to quadrant. ## Magnitude in three dimensions In 3D, the magnitude is $$|ec{A}| = sqrt{A_x^2 + A_y^2 + A_z^2}$$ This appears in forces in space, electric and magnetic fields, fluid flow, and many advanced applications. ## Position, displacement, and velocity vectors A position vector points from the origin to an object's location. A displacement vector describes a change in position: $$Delta ec{r} = ec{r}_f - ec{r}_i$$ Velocity is the rate of change of position. In component form, a velocity vector may be written $$ec{v} = langle v_x, v_y, v_z angle$$ Each component describes motion along one axis. ## Why components matter Components let you turn one vector problem into several simpler one-dimensional problems. A projectile's horizontal motion and vertical motion can be analyzed separately. A force on an incline can be split into components parallel and perpendicular to the surface. Electric fields can be combined component by component. ## The big idea Vectors are the language of direction in physics. Components translate geometric direction into algebra. Magnitude tells the size of a vector, while components tell how it points relative to chosen axes. Mastering vectors is essential for mechanics, electricity, magnetism, waves, and modern physics.