Unit 1 · Topic 1.1 · about 30 minutes · 7 parts · 14 scored checks, 2 of them multiple choice

AP Physics C: Mechanics 1.1: Scalars and Vectors

Q: Should I write a vector in unit vector notation or as a magnitude and a direction?

Both describe the same vector. (5i - 12j) m and 13 m at 67.4° below the +x axis are one displacement written two ways. Use the form the question asks for, and be ready to convert between them.

Q: Where does k come in?

It points along the third axis, +z, which you need to locate a point in space. AP Physics C: Mechanics asks you to analyze motion quantitatively in at most two dimensions, but a position or another vector may still be written with a k component.

Built by Tanner Crow from College Board's course framework · Every answer key independently checked

Unit 1: 10 to 15% of the multiple-choice section

What you'll be able to do

Describe a quantity as a scalar, by its magnitude, or as a vector, by its magnitude and direction, and write a vector in unit vector notation or as a magnitude and a direction.

Key vocabulary

Term	What it means	The exam trap
Scalar	A quantity described by its magnitude only, such as a distance or a speed.	A speedometer reports a scalar, so its reading alone never says which way an object is heading.
Vector	A quantity described by both magnitude and direction, drawn as an arrow whose length is proportional to its magnitude; position, displacement, velocity and acceleration are vectors.	Along one axis a vector's direction is carried by a sign, so state which way is positive before you write one.
Magnitude	The size of a vector, the length of its arrow; for $\vec{r} = A \hat{i} + B \hat{j} + C \hat{k}$ it is $\| \vec{r} \| = \sqrt{A^{2} + B^{2} + C^{2}}$ .	A magnitude is written with bars, $\| \vec{r} \|$ , or as the letter with no arrow, $r$ ; the letter with an arrow means the whole vector.
Unit vector	A vector of magnitude 1 that shows a direction only: $\hat{i}$ , $\hat{j}$ and $\hat{k}$ point along $+ x$ , $+ y$ and $+ z$ , and $\hat{r}$ points along a position vector $\vec{r}$ .	Round its components too far and its magnitude drifts from 1, so keep enough digits that it still works out to 1, and give it no unit.
Position vector	The vector $\vec{r}$ from the origin to a point, written in unit vector notation by its $x$ , $y$ and $z$ components.	A position depends on where you put the origin; a displacement, the change between two positions, does not.
Distance	The length of the path an object follows, a scalar that can only grow as the object moves.	A car's odometer counts distance, so it keeps climbing even when the car doubles back.
Displacement	The change in an object's position, $Δ \vec{r} = {\vec{r}}_{2} - {\vec{r}}_{1}$ , a vector from where it starts to where it ends.	Subtract final minus initial; the other order gives an arrow of the same length pointing the opposite way.
Component	The signed part of a vector along one axis: for $\vec{A} = A_{x} \hat{i} + A_{y} \hat{j}$ , the components are $A_{x}$ and $A_{y}$ .	A vector's components depend on the axes you choose; its magnitude and direction do not.
Resultant vector	The vector sum of two or more vectors, whose components are the sums of their components: $\vec{C} = (A_{x} + B_{x}) \hat{i} + (A_{y} + B_{y}) \hat{j}$ .	Drawn tip to tail, the resultant runs from the first arrow's tail to the last arrow's tip, not from tip to tip.

Tap to open the picture full size

Part 1 Think first How far from the trailhead?

Commit to a guess first. Being wrong now makes the right idea stick better.

Predict first

A hiker walks 9 km due east, then 12 km due north. When she stops, how far is she from where she started, measured in a straight line?

In short. A scalar is a quantity with magnitude only, such as distance or speed; a vector has magnitude and direction, such as position, displacement, velocity or acceleration, and is drawn as an arrow or written in components as $\vec{r} = A \hat{i} + B \hat{j} + C \hat{k}$ .

An arrow's length is proportional to the vector's magnitude, and along a single axis the sign of a component gives its direction.
A vector can be written in unit vector notation or as a magnitude and a direction, and $| \vec{r} | = \sqrt{A^{2} + B^{2} + C^{2}}$ links the two.
To add vectors, add their components: $\vec{C} = (A_{x} + B_{x}) \hat{i} + (A_{y} + B_{y}) \hat{j}$ .

Part 2 Learn it Four ideas, with a check after each

2a. Magnitude only, or magnitude and direction

Some quantities are described completely by one number with a unit. A scalar has a magnitude only: a size, such as 12 m or 5 s. A vector has a magnitude and a direction: 12 m east, or 5 m/s straight up.

Distance and speed are scalars. Distance counts every meter of the path, whichever way the object went, and speed says how fast without saying which way. Position, displacement, velocity and acceleration are vectors. A position says how far and which way a point is from an origin, and a displacement says how far and which way an object ended up from where it started.

One trip can give two different answers. A dog runs 8 m out to a stick and 8 m back: it covers 16 m of distance, but its displacement is zero, because it ends where it began.

A speedometer reads speed, not velocity. Two cars at 25 m/s, one heading north and one heading south, have the same speed and different velocities, because their directions differ.

Quick check · Decide and justify

A runner on a circular track holds a steady 6 m/s for a full lap. Her coach says that because her speed never changes, her velocity never changes either. Is the coach right?

Which reason supports your answer?

2b. Arrows, and signs along a line

A vector is drawn as an arrow. The arrow points in the vector's direction, and its length is proportional to the vector's magnitude. Pick a scale, such as 1 cm for every 5 m/s, and a velocity of 20 m/s becomes an arrow 4 cm long. In the picture at the top of this page one grid square stands for 1 m.

An arrow can be slid anywhere without changing the vector, as long as its length and direction stay the same. That is what lets you place the tail of one arrow at the tip of another, as the picture does.

Along a single line, a direction needs only a sign. Choose which way is positive and say so; the opposite direction is then negative. With east positive, a cart moving 3 m/s west has $v_{x} = - 3$ m/s. The minus sign is the direction, not a smaller amount: $- 3$ m/s and $+ 3$ m/s are arrows of the same length pointing opposite ways. The subscript $x$ says this is the component along the axis, so it needs no arrow over it.

Quick check · Fill the statement

A cart rolls along a straight track at 4 m/s toward the west. Take east as positive, and draw velocity arrows at a scale of 1 cm for every 2 m/s.

The cart's speed is , its velocity is $v_{x} =$ , and its velocity arrow is long.

2c. Components and unit vectors

To write a vector with numbers, split it into components along the axes. The unit vectors $\hat{i}$ , $\hat{j}$ and $\hat{k}$ each have magnitude 1 and point along $+ x$ , $+ y$ and $+ z$ . In unit vector notation a vector is the sum of its components:

\vec{r} = A \hat{i} + B \hat{j} + C \hat{k}

Each of $A$ , $B$ and $C$ is a number with a sign. For a point, $\vec{r}$ is its position vector, the arrow from the origin to the point. With $+ z$ up, a bird at $\vec{r} = (4 \hat{i} - 2 \hat{j} + 4 \hat{k})$ m is 4 m along $+ x$ , 2 m along $- y$ and 4 m up.

The magnitude comes from the Pythagorean theorem, used once for each extra dimension:

| \vec{r} | = \sqrt{A^{2} + B^{2} + C^{2}}

For the bird, $| \vec{r} | = \sqrt{16 + 4 + 16} = 6$ m. Every term under the root is a square, so it is never negative.

Dividing a vector by its own magnitude gives the unit vector in its direction, $\hat{r} = \frac{\vec{r}}{| \vec{r} |}$ . For the bird, $\hat{r} = \frac{2}{3} \hat{i} - \frac{1}{3} \hat{j} + \frac{2}{3} \hat{k}$ . It has magnitude 1 and no unit, so it carries the direction alone.

Quick check · Calculate

A drone hovers at position $\vec{r} = (2 \hat{i} - 3 \hat{j} + 6 \hat{k})$ m from its launch pad, with $+ x$ east, $+ y$ north and $+ z$ up. How far is the drone from the launch pad?

Your answer m

2d. From magnitude and direction to components, and adding

A vector can also be given as a magnitude and a direction, such as 20 m at 30° north of east. To find its components, sketch the arrow and the right triangle under it. With $+ x$ east and $+ y$ north, the side next to the 30° angle lies along $x$ , so

A_{x} = A \cos θ A_{y} = A \sin θ

which gives $(17.3 \hat{i} + 10.0 \hat{j})$ m. That pairing holds only when $θ$ is measured from the $x$ axis; cosine always goes with the side next to the angle. Then give each component the sign of its direction: an arrow pointing north of west has a negative $x$ component.

To add vectors, add their components. The resultant $\vec{C} = \vec{A} + \vec{B}$ is

\vec{C} = (A_{x} + B_{x}) \hat{i} + (A_{y} + B_{y}) \hat{j}

In the picture at the top of this page, $\vec{A} = (3 \hat{i} + 1 \hat{j})$ m and $\vec{B} = (- 1 \hat{i} + 3 \hat{j})$ m, so $\vec{C} = (2 \hat{i} + 4 \hat{j})$ m. Drawn tip to tail, $\vec{C}$ runs from the first tail to the last tip. Its magnitude, $\sqrt{20} \approx 4.5$ m, is less than $| \vec{A} | + | \vec{B} | \approx 6.3$ m, because the two arrows do not point the same way. To get a direction back from components, use $\tan θ = \frac{C_{y}}{C_{x}}$ and read the quadrant from the signs.

Quick check · Match

Take $+ x$ east and $+ y$ north. Tap a displacement given as a magnitude and a direction, then the same displacement in unit vector notation.

Common mistakes

The mistake	Why it is wrong	Do this instead
Adding the magnitudes of vectors that point different ways.	Magnitudes add only when the arrows point the same way. Otherwise part of one arrow cancels the other or runs sideways to it, and the resultant comes out shorter than the sum.	Add the $x$ components, add the $y$ components, and then find the magnitude of the result.
Pairing cosine with the wrong axis, or dropping a component's sign.	Cosine belongs to the side next to the angle, which lies along $y$ when the angle is measured from north or south. A component along $- x$ or $- y$ is negative whatever the trigonometry gives.	Sketch the arrow and its right triangle, mark the side next to the angle, and give each component the sign of its direction.
Reading a negative component as a smaller amount.	Along a line the sign is the direction. A velocity of −5 m/s has a longer arrow than one of +2 m/s.	Compare magnitudes with the signs set aside, and use each sign only to say which way.

Part 3 Watch it done A delivery robot's two legs

Watch each step, then do the last one yourself.

A delivery robot crosses a parking lot. Take $+ x$ east and $+ y$ north. It drives 8.0 m at 30° north of east, then 6.0 m at 45° north of west. What is the magnitude of its resultant displacement?

Step 1 · Resolve the first leg
The 30° is measured from east, the $+ x$ axis, so $A_{x} = 8.0 \cos 30^{\circ} = 6.93$ m and $A_{y} = 8.0 \sin 30^{\circ} = 4.00$ m. That makes $\vec{A} = (6.93 \hat{i} + 4.00 \hat{j})$ m.
Step 2 · Resolve the second leg, with signs
The 45° is measured from west, the $- x$ axis, so the side along $x$ is $6.0 \cos 45^{\circ} = 4.24$ m, pointing west, and the side along $y$ is $6.0 \sin 45^{\circ} = 4.24$ m, pointing north. That makes $\vec{B} = (- 4.24 \hat{i} + 4.24 \hat{j})$ m.
Why this step?
Why is $B_{x}$ negative when $\cos 45^{\circ}$ is positive?
The cosine gives only the size of the side along the $x$ axis. The direction comes from the sketch: the second leg heads partly west, the negative $x$ direction, so its $x$ component carries a minus sign.
Step 3 · Add like components
$\vec{C} = (6.93 - 4.24) \hat{i} + (4.00 + 4.24) \hat{j} = (2.69 \hat{i} + 8.24 \hat{j})$ m. The westward part of the second leg undoes most of the first leg's eastward part, and the two northward parts add.
Your step · Calculate

Finish it. What is the magnitude of the robot's resultant displacement?

Your answer m

Part 4 Try it New situations

Try it · Rank it

Four carts move along the same straight track; take east as positive. Their velocities are: cart A, $v_{x} = - 6$ m/s; cart B, $v_{x} = + 4$ m/s; cart C, $v_{x} = - 2$ m/s; cart D, $v_{x} = + 6$ m/s. Each velocity is drawn as an arrow to the same scale. Rank the arrows by length.

Longest first

Give each one a place, 1 for first. Give two the same place if they are equal.

Cart A
Cart B
Cart C
Cart D

Your ranking: none yet

Try it · Calculate

A camera on a low cable rig over a practice field moves from position ${\vec{r}}_{1} = (4 \hat{i} + 1 \hat{j} + 2 \hat{k})$ m to position ${\vec{r}}_{2} = (7 \hat{i} - 3 \hat{j} + 2 \hat{k})$ m, both measured from one corner of the field with $+ x$ east, $+ y$ north and $+ z$ up. What is the magnitude of the camera's displacement?

Your answer m

Try it · Multiple choice

A displacement $\vec{D}$ has magnitude $D$ and points at an angle $θ$ north of west. Take $+ x$ east and $+ y$ north. Which expression gives $\vec{D}$ in unit vector notation?

On paper. A kayaker paddles 7.0 m at 20° north of east, then 5.0 m at 50° south of east. With a ruler and a protractor, draw both legs tip to tail at 1 cm for every 1 m, draw the resultant from the first tail to the last tip, and measure its length and its angle from east. Then find the resultant by components and compare. A careful drawing lands within a few millimeters and a degree or two of the calculation. There is nothing to submit here.

Part 5 Explain it Draw it, then write it in symbols

Nothing you type leaves this page. Check your answer against the model and each scoring point.

Free response is 50% of the exam score. This one is shaped like Question 2, Translation Between Representations.

Free-response practice · Question 2 style

A student leaves a classroom door and walks a distance $d$ due east down a hallway, then a distance $2 d$ due north. Take $+ x$ east and $+ y$ north, with the origin at the door.

(a) On paper, draw the two displacements tip to tail, with lengths in proportion to their magnitudes, and draw the student's resultant displacement. Label all three arrows.

(b) Write the resultant displacement in unit vector notation, in terms of $d$ .

(c) Derive an expression for the magnitude of the resultant displacement in terms of $d$ , and write the unit vector in its direction.

(d) For $d = 30$ m, calculate the magnitude of the resultant displacement and its direction as an angle north of east.

(e) A classmate says the student ends up $3 d$ from the door. Using your drawing, explain why the classmate is wrong, and name the quantity that does equal $3 d$ .

Your answer stays on this page. It is never sent or saved.

Model answer

(a) An arrow of length $d$ pointing east from the door; from its tip, an arrow twice as long pointing north; and the resultant from the door to the tip of the second arrow, pointing up and to the right, steeper than 45°.

(b) $Δ \vec{r} = d \hat{i} + 2 d \hat{j}$ .

(c) $| Δ \vec{r} | = \sqrt{d^{2} + (2 d)^{2}} = \sqrt{5} d$ . The unit vector is $\frac{Δ \vec{r}}{| Δ \vec{r} |} = \frac{1}{\sqrt{5}} \hat{i} + \frac{2}{\sqrt{5}} \hat{j} \approx 0.45 \hat{i} + 0.89 \hat{j}$ .

(d) $| Δ \vec{r} | = \sqrt{5} (30 m) \approx 67$ m, at $\tan^{- 1} \frac{2 d}{d} = \tan^{- 1} 2 \approx 63^{\circ}$ north of east.

(e) In the drawing, the two legs and the resultant form a right triangle, and the resultant is its hypotenuse. A hypotenuse is shorter than the two legs added together, so the student ends up $\sqrt{5} d \approx 2.24 d$ from the door, not $3 d$ . The $3 d$ is the distance walked, a scalar that adds the lengths of the legs whatever their directions.

Score your own answer. This is a self-check, not a grade. Tick each point your answer makes.

Draws an east arrow and a north arrow twice as long, tip to tail, and the resultant from the first tail to the second tip, all labeled.
Writes the resultant as $d \hat{i} + 2 d \hat{j}$ , with both components positive.
Derives a magnitude of $\sqrt{5} d$ and a unit vector of $\frac{1}{\sqrt{5}} \hat{i} + \frac{2}{\sqrt{5}} \hat{j}$ , with no unit on the unit vector.
Calculates about 67 m at about 63° north of east, with units.
Explains, from the right triangle in the drawing, that the resultant is shorter than $3 d$ , and names $3 d$ as the distance walked.

Part 6 Check yourself A robot's survey log

The case questions use the robot's log. Answer them from that table.

A field robot maps a level field. It starts at the origin; take $+ x$ east and $+ y$ north. Its log lists each leg's displacement and the distance its wheels rolled during that leg, which did not always follow a straight path.

The robot's log
Leg	Displacement (m)	Distance rolled (m)
1	$12 \hat{i} + 5 \hat{j}$	15
2	$- 4 \hat{i} + 10 \hat{j}$	11
3	$- 2 \hat{i} - 7 \hat{j}$	8

Case question 1 · Calculate

Using the log, how far is the robot from its starting point at the end of leg 3?

Your answer m

Case question 2 · Multiple choice

Using the log, which is the unit vector $\hat{r}$ in the direction of the robot's position vector at the end of leg 1?

Case question 3 · Spot the error

A student uses the log to find the robot's position vector at the end of leg 2. Tap the first line with a mistake.

The rest stand on their own and do not use the case.

Check yourself · Sort it

Take $+ x$ east and $+ y$ north, and let $\vec{A} = (3 \hat{i} - 4 \hat{j})$ m. Tap a card, then say whether it describes the same vector as $\vec{A}$ .

Same vector as A

Not the same vector

Check yourself · Decide and justify

A cart rolls along a straight track toward the west at 3 m/s. With east as positive, its velocity is $v_{x} = - 3$ m/s. A student redoes the problem with west as positive. What is the cart's $v_{x}$ now?

Which reason supports your answer?

Check yourself · Put it in order

A problem gives two displacements, each as a magnitude and an angle, and asks for the magnitude and direction of their sum. Put the steps in order.

Part 7 Keep it Hold on to it

There is no earlier lesson in this course to review yet. Starting with 1.2, this part asks two questions from lessons you have already done.

In three lines

A scalar has magnitude only, as distance and speed do; a vector has magnitude and direction, as position, displacement, velocity and acceleration do.
A vector is an arrow whose length is proportional to its magnitude, written as $A \hat{i} + B \hat{j} + C \hat{k}$ or as a magnitude and a direction; along one axis its sign is its direction.
To add vectors, resolve each into components with signs, add like components, and then find the magnitude and direction of the resultant.

Exam strategy for this topic

Section I, Multiple Choice: 42 questions in 85 minutes.
Section II, Free Response: 4 questions in 95 minutes.
This lesson's Explain it is shaped like free-response question 2, Translation Between Representations.
A four-function, scientific, or graphing calculator is allowed on both sections of the exam.

When a question gives a vector as a magnitude and an angle, sketch the arrow and its right triangle before you write a single component. Mark the side next to the angle and the positive direction on each axis. A swapped sine and cosine or a dropped minus sign is easy to make in your head and easy to catch on a sketch.

Questions students ask

Is speed just velocity without the sign?

Along one axis, yes: speed is the magnitude of the velocity, so $v_{x} = - 4$ m/s means a speed of 4 m/s. In two or three dimensions the velocity has several components, and the speed is the magnitude found from all of them.

Should I write a vector in unit vector notation or as a magnitude and a direction?

Both describe the same vector. $(5 \hat{i} - 12 \hat{j})$ m and 13 m at 67.4° below the $+ x$ axis are one displacement written two ways. Use the form the question asks for, and be ready to convert between them.

Where does

\hat{k}

come in?

It points along the third axis, $+ z$ , which you need to locate a point in space. AP Physics C: Mechanics asks you to analyze motion quantitatively in at most two dimensions, but a position or another vector may still be written with a $\hat{k}$ component.

Why does my calculator give the wrong angle?

The inverse tangent returns an angle between −90° and 90°, so it cannot tell $(2 \hat{i} + 3 \hat{j})$ from $(- 2 \hat{i} - 3 \hat{j})$ . Sketch the vector, read its quadrant from the signs of its components, and adjust. Also check that the calculator is in degree mode when your angles are in degrees.

Can a magnitude be negative?

No. A magnitude is a length, so it is zero or positive. Negative signs belong to components, where they show direction.

Course alignment, for teachers

AP Physics C: Mechanics topic 1.1, Unit 1: Kinematics.

Skill 1.A: Create diagrams, tables, charts, or schematics to represent physical situations.
Skill 2.A: Derive a symbolic expression from known quantities by selecting and following a logical mathematical pathway.
Skill 2.B: Calculate or estimate an unknown quantity with units from known quantities, by selecting and following a logical computational pathway.
Skill 3.B: Apply an appropriate law, definition, theoretical relationship, or model to make a claim.

This lesson maps all 8 essential knowledge statements of the topic to the part that teaches each one: 8 are checked by a scored question. The map is in the lesson file.