Topic 1.1: Scalars and Vectors

What Topic 1.1 requires

One learning objective, 1.1.A: describe a scalar or vector quantity using magnitude and direction, as appropriate. Under it sit five essential-knowledge statements, and 1.1.A.4 carries three sub-statements of its own, so eight statements in total.

The CED suggests four skills for this topic: 1.A create diagrams, tables, charts, or schematics to represent physical situations; 2.A derive a symbolic expression from known quantities by selecting and following a logical mathematical pathway; 2.B calculate or estimate an unknown quantity with units from known quantities; and 3.B apply an appropriate law, definition, theoretical relationship, or model to make a claim.

Topic 1.1 prints no boundary statement. Unit 1's three boundary statements sit under Topics 1.3, 1.4 and 1.5, so nothing here fences off how far the vector algebra goes. The fence is elsewhere: the Topic 1.5 boundary statement says AP Physics C: Mechanics only expects students to quantitatively analyze motion in two dimensions, which is why $\hat{k}$ appears in 1.1.A.4.i and then never again in a required calculation.

Unit-vector notation is the whole reason this topic looks different

Statement 1.1.A.4 gives you two legal ways to write a vector, and the CED prints them as equal alternatives: unit vector notation, or a magnitude and a direction. Its relevant equation is

$\hat{i}$, $\hat{j}$ and $\hat{k}$ each have magnitude 1 and no units. They carry direction and nothing else, which is what makes them useful: $A$, $B$ and $C$ are then plain signed numbers with the physical units attached, and every vector operation reduces to arithmetic on those numbers.

Read in that light, a vector in this course is not an arrow you measure with a protractor. It is a triple. That is a small notational change with a large downstream consequence, because a triple can be differentiated one entry at a time. When Topic 1.2 writes $\vec{v} = d\vec{r}/dt$, the operation is defined only because $\vec{r}$ has been written as components first. Every later vector object in the course inherits the same structure: $\vec{F} = d\vec{p}/dt$, $\vec{\tau} = \vec{r} \times \vec{F}$, $\vec{L} = \vec{r} \times \vec{p}$.

Going the other way, from components to a magnitude and a direction, uses geometry the Table of Information supplies in its Geometry and Trigonometry table rather than in its Vectors table. In two dimensions,

and in three, $\left| \vec{A} \right| = \sqrt{A_x^2 + A_y^2 + A_z^2}$. Neither of those is printed in the Vectors table. The Pythagorean relation $a^2 + b^2 = c^2$ and the three ratios $\sin\theta = a/c$, $\cos\theta = b/c$, $\tan\theta = a/b$ are printed, in the Geometry and Trigonometry table, next to a right-triangle figure.

The inverse tangent has a sign problem the CED does not warn you about. $\tan^{-1}(A_y/A_x)$ returns the same value for $(3, 4)$ and for $(-3, -4)$, which point in opposite directions. Read the signs of the two components to decide the quadrant, then state the direction relative to a named axis.

Statement 1.1.A.4.ii: the position vector and the unit vector along it

This is the shortest statement in the topic and the one students skip. The position vector of a point is $\vec{r}$, and the unit vector in the direction of the position vector is denoted $\hat{r}$.

The relationship the notation implies is

Dividing a vector by its own magnitude strips the units and the size and leaves the direction. Nothing in Unit 1 forces you to use $\hat{r}$, which is why it is easy to file away as decoration. It becomes load-bearing in Unit 2 and again across the whole of AP Physics C: Electricity and Magnetism, wherever a law points along the line joining two things: a gravitational or Coulomb force, an inverse-square field, a Biot-Savart element. In each case the magnitude is a scalar expression and $\hat{r}$ supplies the direction, so the vector statement is the product of the two.

Worth noticing: $\hat{r}$ is not a fixed direction. It changes as the point moves, which is exactly why circular motion has a centripetal acceleration even at constant speed. A hat means unit length, not constant.

The Vectors table on the Table of Information, all five lines

The AP Physics C sheet is not one table. The Table of Information also carries separate Vectors, Calculus, Geometry and Trigonometry, and Identities tables, and Topic 1.1 draws on the first of those. The Vectors table prints exactly five lines. Here they are, in the order the booklet prints them:

Three of Topic 1.1's own printed equations are in that list: the unit-vector expansion at 1.1.A.4.i, and the two resultant forms at 1.1.A.4.iii. The other two lines, the dot and cross product magnitudes, belong to Topic 1.1's subject matter but are not printed anywhere in Unit 1's required content. They arrive later, and the sheet gives them to you early.

Count the components in the last line: it stops at $\hat{j}$. Component addition is printed in two dimensions only, matching the Topic 1.5 boundary statement, even though the line above it expands a vector in three. If a question hands you three-component vectors to add, extend the pattern yourself; the sheet will not do it for you.

One more thing the Vectors table does not print: the component forms $\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z$ and the determinant form of the cross product. Only the magnitude expressions in terms of the angle between the vectors are printed.

Dot and cross, and why one is a number and the other is a vector

Both printed lines involve two vectors and an angle, and they are the two ways to multiply vectors that physics needs. Which one a situation calls for is decided by what kind of answer the situation should have.

Notice the deliberate asymmetry in how the sheet prints them. The dot product line is a full equation, because a scalar needs no direction. The cross product line is wrapped in absolute-value bars, because it gives you the magnitude only: the direction has to come from a right-hand rule, which no table can print. That is worth remembering as a reading skill. When a printed equation carries magnitude bars, the sheet is telling you it has withheld the direction.

One identity worth carrying, though it is nowhere in the CED, is that the two products divide the product of the magnitudes between them:

because $\cos^2\theta + \sin^2\theta = 1$, and that identity is printed, in the Identities table. It is a fast arithmetic check on a pair of answers.

Adding by components, and what a resultant is not

Statement 1.1.A.4.iii defines the resultant as the vector sum of the addend vectors' components, and prints both a compact form and an explicit one:

The second is the procedure and the first is the claim. Add the $\hat{i}$ parts, add the $\hat{j}$ parts, and never mix them, because $\hat{i}$ and $\hat{j}$ are perpendicular and no amount of one contributes to the other. That independence is the same fact Topic 1.5 states as physics at 1.5.A.3.

What a resultant is not is the sum of the magnitudes. $\left| \vec{A} + \vec{B} \right|$ equals $A + B$ only when the two vectors are parallel and point the same way; it equals $\left| A - B \right|$ when they are antiparallel, and lands somewhere between for every other angle. Adding magnitudes is the single most common error on this topic and it is easy to catch, because it always produces a number that is too large.

Subtraction follows from 1.1.A.5, which says opposite directions are denoted by opposite signs: $\vec{A} - \vec{B} = \vec{A} + \left( -\vec{B} \right)$, and negating a vector negates every component. That is the operation Topic 1.4 needs for relative velocity.

One dimension, where the sign carries the direction

Statement 1.1.A.5 is the whole of AP Physics 1's Topic 1.1 compressed into one line: in a given one-dimensional coordinate system, opposite directions are denoted by opposite signs.

The two words doing the work are given and a. A sign is meaningful only against a coordinate system somebody has declared, and it has no meaning at all until then. "The acceleration is $-6.0$ m/s$^2$" is a complete statement once you have said which way is positive and not before. On a free-response question, drawing and labelling the axis is part of the answer, and the AP Physics C: Mechanics exam allocates 20 to 35% of the free-response score to Practice 1, creating representations.

A sign is also not a size. $-9$ m/s is a faster motion than $+4$ m/s. Compare magnitudes when the question asks about speed, and compare signed values when it asks about direction or about which way a quantity is changing.

The C course drops one sub-statement AP Physics 1 prints here. AP Physics 1's 1.1.A.3.i says vectors are notated with an arrow above the symbol, and its 1.1.A.3.ii says vector notation is not required for vector components along an axis. Neither appears in the Physics C framework. In practice the C course uses the arrow throughout, and it uses subscripted components without arrows, exactly as Physics 1 describes. The convention did not change; the CED simply stopped spelling it out, because a course that writes $\vec{r} = A\hat{i} + B\hat{j} + C\hat{k}$ has already shown you both notations at once.

Traps on this topic

Adding magnitudes instead of components. Covered above, and worth a second mention because it survives into Unit 2 as adding force magnitudes to get a net force.

Treating $\hat{i}$, $\hat{j}$, $\hat{k}$ as if they had units. They do not. In $\vec{r} = 6.0\hat{i} - 3.0\hat{j}$ metres, the metres belong to the coefficients. Writing "$6.0\hat{i}$ m" and "6.0 m in the $x$ direction" are the same statement.

Losing the quadrant to the inverse tangent. A calculator returns an angle between $-90^\circ$ and $+90^\circ$. Two of the four quadrants are not in that range.

Calling a negative quantity a decreasing one. $-6.0$ m/s$^2$ is an acceleration pointing in the negative direction. Whether the object is speeding up depends on the direction of its velocity, not on the sign of its acceleration alone. The CED names this misconception in Unit 1's "Building the Science Practices" page, where it says practice with multiple representations helps dispel common misconceptions about motion, such as exclusively using negative acceleration to describe an object slowing down.

Assuming a unit vector is a constant. $\hat{r}$ points from the origin toward a moving point, so it rotates as the point moves. Differentiating a vector written as (magnitude) times $\hat{r}$ requires differentiating both factors.

Reaching for the cross product's direction from the printed line. The sheet gives $AB\sin\theta$ inside magnitude bars. The direction is a right-hand-rule question every time.

If you want the algebra-based version of this topic

Both courses have a Topic 1.1, and this is one of the four C: Mechanics topics whose title differs from its AP Physics 1 counterpart. That difference is real rather than cosmetic.

The pattern is a clean swap. AP Physics 1 spends a whole learning objective on adding vectors in one dimension and prints a kinematic equation to make the point about arrow notation. AP Physics C compresses one-dimensional signs into a single closing statement and spends the topic on unit vectors and components instead. The two skill lists differ too: the C version adds 2.A, derive a symbolic expression, and 2.B, calculate an unknown quantity, and drops 3.C, justify a claim using evidence.

If you are studying for AP Physics 1, the page you want is Scalars and Vectors in One Dimension, and the scalar versus vector comparison serves both courses. If you are studying for AP Physics C: Mechanics, stay here, then read Topic 1.2, which differentiates the position vector you just learned to write.