AP Physics 1

Scalars and vectors in one dimension

A scalar quantity is described by a magnitude alone: a number and a unit. A vector quantity needs a magnitude and a direction. Distance, speed, time, mass, energy and power are scalars. Position, displacement, velocity, acceleration, force and momentum are vectors.

A vector is drawn as an arrow. The arrow points in the direction of the quantity and its length is proportional to the magnitude, so a velocity arrow twice as long stands for twice the speed. In print a vector is written with an arrow over the symbol, as in $\vec{v}$, and the symbol without the arrow usually means the magnitude.

In one dimension the direction of a vector is carried entirely by its sign. A coordinate axis is chosen, one direction along it is called positive, and every vector component along the axis is then a positive or negative number. A car moving at $20\ \mathrm{m/s}$ to the right and a car moving at $20\ \mathrm{m/s}$ to the left have the same speed and opposite velocities, $v_x = +20\ \mathrm{m/s}$ and $v_x = -20\ \mathrm{m/s}$. The choice of positive direction is free, but it must be kept for every quantity in a problem. If up is positive for velocity, then up is positive for acceleration, force and displacement as well.

Adding vectors in one dimension

Vectors along the same line are added by adding their signed components. Opposite directions have opposite signs, so they partly or wholly cancel. A boat that can move at $6\ \mathrm{m/s}$ relative to the water, heading upstream against a current of $2\ \mathrm{m/s}$, has a velocity relative to the bank of $+2 + (-6)$ taken with downstream positive, which is $-4\ \mathrm{m/s}$, meaning $4\ \mathrm{m/s}$ upstream.

The result of adding vectors is called the vector sum, or the net or resultant vector. The same rule gives the net force on an object when forces act along one line: a $30\ \mathrm{N}$ push to the right and a $12\ \mathrm{N}$ push to the left give a net force of $+18\ \mathrm{N}$.

Vectors in two dimensions

A vector in a plane can be replaced by two perpendicular components, one along each of two chosen axes, so that the vector is the resultant of its components. Adding the components back together reproduces the original vector. The axes are usually horizontal and vertical, but any perpendicular pair may be chosen, and a good choice makes a problem much easier. On an incline, for instance, one axis is parallel to the slope.

Resolving a vector

If a vector of magnitude $A$ makes an angle $\theta$ with the $x$ axis, the right-angled triangle formed by the vector and its components gives

where the component along the axis the angle is measured from uses cosine, and the component across it uses sine. The components can be positive or negative depending on the quadrant. The magnitude and angle are recovered from the components with

The first is Pythagoras' theorem, and the second gives the angle measured from the $x$ axis, for which the signs of the components say which quadrant the vector is in.

A worked example: a person pulls a sled with a rope that makes $30°$ above the horizontal with a force of $80\ \mathrm{N}$. The horizontal component is $80\cos 30° = 69\ \mathrm{N}$ and the vertical component is $80\sin 30° = 40\ \mathrm{N}$. Only the horizontal part pulls the sled along the ground, and the vertical part lifts slightly on the sled.

Adding vectors with components

To add vectors in two dimensions, resolve each along the same axes, add all the $x$ components to get $R_x$, add all the $y$ components to get $R_y$, and then combine these into a magnitude and angle.

Take two displacements: $3.0\ \mathrm{m}$ east followed by $4.0\ \mathrm{m}$ north. With east as $+x$ and north as $+y$ the components of the sum are $R_x = 3.0\ \mathrm{m}$ and $R_y = 4.0\ \mathrm{m}$, so

north of east. The distance travelled is $7.0\ \mathrm{m}$ but the displacement has magnitude $5.0\ \mathrm{m}$, which is why a vector sum is not the sum of the magnitudes. The magnitude of the sum equals the sum of the magnitudes only when the vectors point the same way.

A vector sum is independent of the axes

The two components of a vector depend on the axes chosen, but the vector does not. Rotating the axes changes $A_x$ and $A_y$ and leaves $A$ unchanged. Perpendicular components are independent of each other in a useful sense: a force or an acceleration along one axis changes motion along that axis only. This is what allows two-dimensional motion to be split into two one-dimensional problems, which is the method used for projectiles, inclines and circular motion.