Topic 1.1 Scalars and Vectors: describe scalar and vector quantities by magnitude and direction, resolve a vector into perpendicular components, and add vectors by components and graphically.

What this topic is asking

The College Board (Topic 1.1) wants you to describe physical quantities as either scalars or vectors, to resolve a vector into perpendicular components using sine and cosine, and to add vectors both graphically and by components. In AP Physics C: Mechanics, which is calculus-based, vectors are the language of everything that follows: position, velocity, acceleration, force, momentum and angular momentum are all vectors, and the kinematics that comes next defines velocity and acceleration as derivatives of vector quantities. Getting the vector toolkit right is the foundation for the whole course.

Scalars versus vectors

The distinction matters because vectors do not add like ordinary numbers. Walking $3$ m east then $4$ m north leaves you $5$ m from the start, not $7$ m, because the displacements point in different directions. Speed (a scalar) is the magnitude of velocity (a vector); distance (a scalar) is the path length, while displacement (a vector) is the straight-line change in position. Throughout AP Physics C you keep this separation in mind: energy and work are scalars, but force and momentum are vectors that must be resolved.

Resolving a vector into components

Any vector in a plane can be written as the sum of two perpendicular components. If $\vec{A}$ has magnitude $A$ and makes angle $\theta$ with the positive $x$-axis, then

Going the other way, the magnitude and direction follow from the components:

The catch with the inverse tangent is that it returns an angle only in two of the four quadrants, so always check the signs of $A_x$ and $A_y$ to place the vector correctly. A vector pointing up and to the left has $A_x < 0$ and $A_y > 0$ (second quadrant), even though the calculator may report a first- or fourth-quadrant angle.

Unit-vector notation

It is often cleanest to write a vector in terms of the unit vectors $\hat{\imath}$ (pointing along $+x$) and $\hat{\jmath}$ (pointing along $+y$), each of magnitude one:

Adding vectors then reduces to adding like components: $\vec{A} + \vec{B} = (A_x + B_x)\hat{\imath} + (A_y + B_y)\hat{\jmath}$. A unit vector in the direction of $\vec{A}$ is $\hat{A} = \vec{A}/|\vec{A}|$, which strips out the magnitude and keeps only the direction. This notation becomes essential in AP Physics C when you differentiate a position vector $\vec{r}(t) = x(t)\hat{\imath} + y(t)\hat{\jmath}$ component by component to get velocity and acceleration.

Adding vectors graphically

Graphically, you add vectors tip to tail: draw the first vector, start the second at the tip of the first, and the resultant runs from the tail of the first to the tip of the last. Equivalently, the parallelogram rule places both vectors tail to tail and the resultant is the diagonal. Subtraction $\vec{A} - \vec{B}$ is the addition $\vec{A} + (-\vec{B})$, where $-\vec{B}$ has the same magnitude as $\vec{B}$ but the opposite direction. The graphical picture is useful for a quick sanity check, but the component method is what you compute with.

Try this

Q1. A vector has components $A_x = -3.0$ and $A_y = 4.0$. Calculate its magnitude and the angle it makes with the positive $x$-axis. [3 points]

• Cue. $A = \sqrt{(-3.0)^2 + 4.0^2} = 5.0$. The reference angle is $\tan^{-1}(4.0/3.0) = 53^\circ$; since $A_x < 0$, $A_y > 0$ (second quadrant), the direction is $180^\circ - 53^\circ = 127^\circ$.

Q2. Write the unit vector in the direction of $\vec{A} = 6.0\,\hat{\imath} + 8.0\,\hat{\jmath}$. [2 points]

• Cue. $|\vec{A}| = 10$, so $\hat{A} = 0.60\,\hat{\imath} + 0.80\,\hat{\jmath}$.