AP Physics C Vectors Box: Dot and Cross Products

The Box as the Booklet Prints It

The box is headed VECTORS and it is the leftmost of three boxes on the final appendix page of each Physics C booklet, sharing a row with CALCULUS and IDENTITIES. It holds five lines and nothing else.

Five lines. Two products, one unit-vector expansion, and two statements of addition.

Where it is printed:

• AP Physics C: Mechanics, appendix page 206
• AP Physics C: Electricity and Magnetism, appendix page 181

The two printings are identical. Neither AP Physics 1 nor AP Physics 2 prints a vectors box. Of the five mathematics tables, their booklets carry the geometry box and the trig values table and no others, which is consistent with the algebra-based courses handling vectors by drawing triangles rather than by taking products.

Yes, the Cross Product Magnitude Is Printed

Line 2 deserves its own section, because it is the single equation most often reported as missing from the AP sheet:

It is printed. It sits directly under the dot product in the VECTORS box on appendix page 206 of the C: Mechanics CED and appendix page 181 of the C: E&M CED, both Effective Fall 2024. This page's transcription of it was taken from a rendered image of those pages at 400 dots per inch, not from any secondary source.

The reason it gets reported as missing is worth understanding, because the same failure will mislead you about four other tables. Most machine-readable transcriptions of the AP equation sheet, including this site's own, cover the physics equations and the constants boxes only, and explicitly leave out the geometry, trigonometry, vector, calculus and identity tables. So a search of such a transcription returns nothing for the cross product, and the natural conclusion is that the exam does not print it. That conclusion is wrong. Absence from a transcription is not absence from the booklet.

What the line gives you is the magnitude only. The direction of $\vec{A} \times \vec{B}$ is not printed anywhere, and neither is the right-hand rule that fixes it. See the omissions section below.

Dot or Cross: Which One the Sheet Wants

The two products are told apart by their trig function, and the trig function is told apart by geometry. The dot product asks how much of one vector lies along the other, so it peaks when they are parallel and vanishes at right angles: that is a cosine. The cross product asks how much of one vector lies across the other, so it vanishes when they are parallel and peaks at right angles: that is a sine.

Once you can name which product a quantity is, you know which trig function belongs in it without looking anything up. The sheet's own equations sort cleanly into the two families.

Quantities the C sheets write as dot products, including work $W = \int \vec{F}\cdot d\vec{r}$, potential energy change $\Delta U = -\int_a^b \vec{F}_{cf}(r)\cdot d\vec{r}$, rotational work $W = \int \tau \cdot d\theta$, which the booklet prints without vector arrows, electric flux $\Phi_E = \int \vec{E}\cdot d\vec{A}$, Gauss's law $\oint \vec{E}\cdot d\vec{A} = q_{enc}/\varepsilon_0$, potential difference $\Delta V = -\int \vec{E}\cdot d\vec{r}$, current from current density $I = \int \vec{J}\cdot d\vec{A}$, magnetic flux $\Phi_B = \int \vec{B}\cdot d\vec{A}$, and Ampere's law $\oint \vec{B}\cdot d\vec{\ell} = \mu_0 I_{enc}$.

Quantities the C sheets write as cross products, including torque $\vec{\tau} = \vec{r} \times \vec{F}$, angular momentum $\vec{L} = \vec{r} \times \vec{p}$, the magnetic force on a moving charge $\vec{F}_B = q\left(\vec{v} \times \vec{B}\right)$, the Biot-Savart law $d\vec{B} = \frac{\mu_0}{4\pi}\frac{I\left(d\vec{\ell} \times \hat{r}\right)}{r^2}$, and the force on a current-carrying wire $\vec{F}_B = \int I\left(d\vec{\ell} \times \vec{B}\right)$.

The algebra-based sheets state the same physics without the vector notation, printing $W = Fd\cos\theta$ where the C sheet prints a dot product and $\tau = rF\sin\theta$ and $F_B = qvB\sin\theta$ where the C sheet prints a cross product. The sine and the cosine are already there in the algebra-based forms; the C booklet just names where they come from.

The Notation Quirk in Line 3

Line 3 reads $\vec{r} = \left(A\hat{i} + B\hat{j} + C\hat{k}\right)$, and lines 4 and 5 read $\vec{C} = \vec{A} + \vec{B}$ and $\vec{C} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}$.

Read those three lines together and you will notice that $A$, $B$ and $C$ change meaning between them. In line 3 they are scalar components: $A$ is the number multiplying $\hat{i}$. In lines 4 and 5 they are the names of vectors, so $A_x$ is the $x$ component of the vector $\vec{A}$ and $C$ is a vector with an arrow over it. The letter $C$ in particular is a scalar $z$ component on one line and a vector two lines later.

This is a genuine quirk of the printed box, not a transcription slip on our part, and it is repeated identically in both Physics C booklets. It is harmless once you have seen it, and confusing if you have not, so it is worth reading once before exam day rather than during.

A second asymmetry in the same three lines: line 3 is three-dimensional and line 5 is only two-dimensional. The unit-vector expansion carries $\hat{i}$, $\hat{j}$ and $\hat{k}$, but the component-addition rule stops at $\hat{i}$ and $\hat{j}$. The $z$ component adds exactly as you would expect, and the booklet simply does not print that case. Do not read the omission as a restriction: C: Mechanics states topic 1.5 as motion in two or three dimensions, and three-dimensional vectors are fair game.

What the Box Does Not Give You

The vectors box is five lines, and the gaps are larger than the content. Every item below is assumed knowledge, printed nowhere in either Physics C booklet:

• The magnitude of a vector from its components. There is no $A = \sqrt{A_x^2 + A_y^2}$ on any AP sheet. What you get instead is $a^2 + b^2 = c^2$ in the geometry box, which is the same statement about a triangle. You supply the connection.
• The components of a vector from its magnitude and angle. No $A_x = A\cos\theta$ and no $A_y = A\sin\theta$. Again the geometry box gives $\cos\theta = b/c$ and $\sin\theta = a/c$, and again you rearrange.
• The dot product in component form. No $\vec{A}\cdot\vec{B} = A_xB_x + A_yB_y + A_zB_z$. The box prints only the geometric definition, so a question giving you two vectors in unit-vector notation expects you to know the component form or to find the angle first.
• The cross product in component form, and no determinant layout. Line 2 gives you a magnitude and nothing about the resulting vector's components.
• The direction of a cross product. No right-hand rule, no statement that $\vec{A}\times\vec{B}$ is perpendicular to both, and no note that $\vec{B}\times\vec{A} = -\vec{A}\times\vec{B}$. Every direction in every torque, angular momentum and magnetic force question comes from a rule that is not printed.
• The unit vector of a given vector, so no $\hat{A} = \vec{A}/A$, even though $\hat{r}$ appears inside the Biot-Savart law and the point-charge field on the E&M sheet.
• Products of the unit vectors themselves, so no $\hat{i}\cdot\hat{i} = 1$ and no $\hat{i}\times\hat{j} = \hat{k}$.
• Vector subtraction, though line 4 plus a minus sign covers it.
• The scalar triple product or the vector triple product.
• Any statement about angles beyond $\theta$ being the angle between the two vectors, which the box does not define in words either.

The pattern is consistent: the booklet prints the definitions that carry a physical meaning and leaves the mechanical component algebra to you. That is a reasonable division, but it means the vectors box is not a substitute for knowing how to resolve a vector.

The Right-Hand Rule, Which You Have to Supply

Since the direction of a cross product is the part the booklet omits, here is the piece you have to bring.

For $\vec{C} = \vec{A} \times \vec{B}$: point the fingers of your right hand along $\vec{A}$, curl them toward $\vec{B}$ through the smaller angle, and your thumb points along $\vec{C}$. The result is perpendicular to both $\vec{A}$ and $\vec{B}$, so it is perpendicular to the plane they define.

Three consequences that show up constantly:

• Order matters. $\vec{B} \times \vec{A}$ points the opposite way to $\vec{A} \times \vec{B}$. Writing $\vec{\tau} = \vec{F} \times \vec{r}$ instead of $\vec{r} \times \vec{F}$ reverses your torque, which reverses the sense of rotation you predict.
• The magnitude vanishes when the vectors line up, which line 2 does tell you through $\sin 0° = 0$. A force applied along the radius produces no torque; a charge moving parallel to a magnetic field feels no magnetic force.
• Negative charges reverse the force but not the cross product. In $\vec{F}_B = q(\vec{v} \times \vec{B})$, the cross product itself is fixed by the geometry, and the sign of $q$ then flips the force. Do the geometry first and apply the sign last.

For the flux integrals the same care applies to $d\vec{A}$, whose direction is the outward normal for a closed surface. The sheet writes $\oint \vec{E}\cdot d\vec{A}$ and relies on you knowing which way $d\vec{A}$ points, because that is what makes the flux out of a closed surface positive.

How the Two Products Fit Together

There is a neat relation between the two printed lines that follows immediately from an identity in the box next door, and it makes a good self-check on any pair of numbers you compute.

Square both printed products and add them:

The last step used $\sin^2\theta + \cos^2\theta = 1$, which is printed in the IDENTITIES box on the same page. So the dot product and the cross-product magnitude are the two legs of a right triangle whose hypotenuse is $AB$: whatever one product loses as the angle changes, the other gains.

That gives you two practical checks. First, neither product can exceed $AB$, so a dot product larger than the product of the magnitudes is an arithmetic error. Second, if a question gives you both products for the same pair of vectors, you can find $AB$ without knowing either magnitude, and then $\tan\theta$ is the cross-product magnitude divided by the dot product.

The same relation explains why a force perpendicular to a displacement does no work but exerts maximum torque about the pivot: those are the two ends of the same triangle.

Physics 1 and Physics 2 Candidates

Neither algebra-based booklet prints a vectors box, and neither exam requires the dot or cross product as such. What you get instead, in exchange, is the same physics written out with the trig function already inserted:

The magnetic flux row is the exception that proves the rule: AP Physics 2 does print a dot product, in $\Phi_B = \vec{B}\cdot\vec{A}$, and it prints the expanded form $\Phi_B = \lvert\vec{B}\rvert\cos\theta\,\lvert\vec{A}\rvert$ immediately beneath it, which is the definition spelled out for a course with no vectors box.

So an algebra-based candidate is not missing the concept, only the notation. If you are moving up to Physics C, the translation is short: a $\cos\theta$ in an algebra-based formula is a dot product waiting to happen, and a $\sin\theta$ is a cross product.