Scalars, Vectors, and Coordinate Systems

Describing quantities that have both magnitude and direction, and locating objects precisely in space

You give a friend directions to your house. “Turn left, drive 400 meters” gets her there. “Drive 400 meters” does not — because direction is missing. Physics faces the same problem. A speedometer reading of 60 km/h tells you how fast you are moving; it does not tell you which way. To describe motion, forces, and fields completely, physicists use a mathematical object that encodes both magnitude and direction. These objects are called vectors, and mastering them is essential before you can understand kinematics, Newton’s laws, or projectile motion.

4.1 Why Direction Matters in Physics

Many of the most important quantities in physics have a direction component that fundamentally changes the physical situation. Consider a force pushing a box: a force of 50 N to the right produces a different outcome than a force of 50 N upward, even though the magnitude is the same. Consider two displacements: walking 3 km north and then 4 km east brings you to a different place than walking 3 km south and then 4 km east, even though the total distances walked are identical.

Direction is not a minor detail; it is often the most important part of the description. A ball thrown upward and a ball thrown downward both start with the same speed, but their subsequent motions are completely different. A current flowing into a junction and a current flowing out of the same junction have opposite physical effects. In electromagnetism, the direction of a magnetic force on a moving charge is perpendicular to both the charge’s velocity and the magnetic field — a result that cannot be expressed without direction.

Physics recognizes this by dividing all physical quantities into two categories: scalars, which require only a number and a unit, and vectors, which require a number, a unit, and a direction. Understanding this distinction is the first step to describing the physical world precisely.

4.2 Scalar and Vector Quantities

A scalar quantity is fully described by a magnitude — a single number with a unit. Scalars have no direction associated with them. They are added, subtracted, multiplied, and divided by ordinary arithmetic.

A vector quantity requires both a magnitude and a direction to be fully specified. Two vectors with the same magnitude but different directions are not equal — they represent physically different situations.

Table 4.1 Scalar and vector quantities compared. Note that each vector pair has a scalar counterpart that carries magnitude without direction.

Key Point: Speed is the scalar “how fast.” Velocity is the vector “how fast and which way.” Distance is the scalar total path length. Displacement is the vector change in position. These distinctions become critical in Chapters 6–13.

4.3 Magnitude and Direction

The magnitude of a vector is its size, expressed as a positive number with a unit. Magnitude is always non-negative; it measures how large the vector quantity is without regard to direction. The magnitude of a velocity vector of 15 m/s north is 15 m/s. The magnitude of a force vector of 80 N downward is 80 N.

Direction can be expressed in several ways, depending on the context. In everyday language, compass directions (north, south, east, west) work well for horizontal motion. In physics problems, angles measured from a reference direction (usually the positive x-axis) provide a precise, unambiguous description. In one-dimensional problems, a positive or negative sign often suffices: positive means one direction, negative means the opposite.

Table 4.2 Common ways to express vector direction. Component notation is introduced in Section 4.7.

4.4 Representing Vectors with Arrows

The most natural representation of a vector is an arrow. The length of the arrow represents the magnitude of the vector: a longer arrow means a larger quantity. The direction the arrow points represents the direction of the vector quantity.

In diagrams, vectors are drawn to scale: if a force of 10 N is represented by an arrow 2 cm long, then a force of 20 N is represented by an arrow 4 cm long. The tail of the arrow is the starting point; the head (the pointed end) is the terminal point and indicates the direction.

Tip: When drawing vectors, always label the arrow with the symbol for the quantity (v, F, a) and its magnitude and unit. Include a direction label or compass rose so the reader knows the reference frame. A vector arrow without labels is incomplete.

The same vector can be drawn starting from any point — an arrow’s position on the page does not change its value. What matters is its length and direction. Two arrows of the same length pointing the same way represent the same vector, regardless of where they are drawn on the diagram.

Key Point: A vector is defined entirely by its magnitude and direction. Moving it parallel to itself (without rotating it or stretching it) produces the same vector. This property is called translational invariance and is used when adding vectors graphically.

4.5 Vector Notation and Labels

Physicists use specific notation to distinguish vectors from scalars in written work. In printed textbooks, vectors are typically written in boldface: “v” (scalar speed) versus “v” (vector velocity). When writing by hand, draw a small arrow over the symbol: ⃗v for a vector, v for a scalar.

Printed: v (scalar) v (vector)

Handwritten: v (scalar) v⃗ (vector)

The magnitude of a vector “v” is written as |v| or simply as the unbolded symbol v. For example, if a velocity vector v⃗ has magnitude 12 m/s and points north, we write |v⃗| = 12 m/s, or just v = 12 m/s when the context makes clear that we are referring to the magnitude only.

Common Mistake: In this book, vector symbols are marked with an overhead arrow in handwritten work. In diagrams, always include an arrowhead at the tip of a vector. Never use an arrowhead on a scalar line segment.

Components of a vector are labeled with subscripts. If a vector F⃗ acts in two dimensions, its horizontal component is written Fₓ and its vertical component is Fᵧ (or Fₕ). These are signed scalar quantities, introduced in Section 4.7.

4.6 Adding and Subtracting Vectors Graphically

Vectors are added differently from scalars. You cannot simply add magnitudes unless the vectors point in exactly the same direction. The correct graphical method is called the tip-to-tail method (also known as the triangle method or the polygon method).

The Tip-to-Tail Method for Vector Addition

Step 1: Draw the first vector as an arrow from its tail to its tip.

Step 2: Place the tail of the second vector at the tip of the first vector.

Step 3: Draw the resultant vector from the tail of the first vector to the tip of the second vector.

Step 4: Measure the length and direction of the resultant to find its magnitude and direction.

Example — Walking north then east Walk 3.0 km north, then 4.0 km east. The resultant displacement has magnitude √(3² + 4²) = 5.0 km, at angle θ = arctan(4/3) = 53° east of north.

The order in which vectors are added does not matter. The resultant of A⃗ + B⃗ is the same as B⃗ + A⃗. This is the commutative property of vector addition. When more than two vectors are added, they are placed tip-to-tail in sequence; the resultant runs from the tail of the first to the tip of the last.

The Parallelogram Method

An alternative graphical method for adding two vectors: place both vectors tail-to-tail. Construct a parallelogram using the two vectors as adjacent sides. The diagonal of the parallelogram, starting from the shared tail, is the resultant vector. Both methods give the same result.

Vector Subtraction

To subtract vector B⃗ from vector A⃗, add the negative of B⃗. The negative of a vector has the same magnitude but opposite direction. So A⃗ − B⃗ = A⃗ + (−B⃗). Graphically, reverse the arrow of B⃗ and then apply the tip-to-tail method.

A⃗ − B⃗ = A⃗ + (−B⃗)

Common Mistake: The magnitude of A⃗ + B⃗ is not generally equal to |A⃗| + |B⃗|. That would only be true if both vectors pointed in exactly the same direction. In all other cases, the magnitude of the resultant is less than the sum of the individual magnitudes.

4.7 Vector Components

Graphical vector addition gives approximate answers and is hard to apply to three-dimensional problems. A more powerful method breaks each vector into components along perpendicular axes, works with scalars along each axis separately, and then reconstructs the resultant at the end.

The components of a vector are the projections of that vector onto the axes of a coordinate system. For a vector in two dimensions, there is one component along each axis. Components are signed scalar quantities: they can be positive (pointing in the positive axis direction) or negative (pointing in the negative axis direction).

Aₓ = |A⃗| cos θ Aᵧ = |A⃗| sin θ

where θ is the angle measured counterclockwise from the positive x-axis to the vector.

Conversely, given the components, the magnitude and direction can be recovered:

|A⃗| = √(Aₓ² + Aᵧ²) θ = arctan(Aᵧ / Aₓ)

Use the signs of Aₓ and Aᵧ to determine the correct quadrant for θ.

Tip: The component equations use cosine for the x-axis and sine for the y-axis when θ is measured from the positive x-axis. If θ is measured from the positive y-axis (as in some navigation problems), swap cos and sin. Always draw a diagram and identify which angle you are using.

4.8 Resolving Vectors into Horizontal and Vertical Parts

Resolving a vector means finding its components along chosen perpendicular directions. In most introductory physics problems, these directions are horizontal (x) and vertical (y), because gravity acts vertically and many forces act horizontally.

Worked Example 4.1 — Resolving a Force

Problem: A rope pulls a sled with a force of 80.0 N at an angle of 30.0° above the horizontal. Find the horizontal and vertical components.

Step 1: Identify angle: 30.0° above horizontal → measured from positive x-axis: θ = 30.0°

Step 2: x-component: Fₓ = 80.0 N × cos(30.0°) = 80.0 × 0.866 = 69.3 N (rightward, positive)

Step 3: y-component: Fᵧ = 80.0 N × sin(30.0°) = 80.0 × 0.500 = 40.0 N (upward, positive)

Step 4: Check: |√(69.3² + 40.0²)| = √(4802 + 1600) = √6402 ≈ 80.0 N ✔

Worked Example 4.2 — Adding Vectors by Components

Problem: Vector A⃗ = 40.0 N at 0°; Vector B⃗ = 30.0 N at 90°. Find the resultant.

Step 1: Components of A⃗: Aₓ = 40.0 N, Aᵧ = 0.0 N

Step 2: Components of B⃗: Bₓ = 0.0 N, Bᵧ = 30.0 N

Step 3: Add components: Rₓ = 40.0 + 0.0 = 40.0 N; Rᵧ = 0.0 + 30.0 = 30.0 N

Step 4: Resultant: |⃗R| = √(40.0² + 30.0²) = √2500 = 50.0 N; θ = arctan(30.0/40.0) = 36.9° above horizontal

The General Component Method for Adding Vectors

• Resolve every vector into x and y components using sine and cosine.

• Add all x-components to get the resultant x-component: Rₓ = Aₓ + Bₓ + Cₓ + …

• Add all y-components to get the resultant y-component: Rᵧ = Aᵧ + Bᵧ + Cᵧ + …

• Find the resultant magnitude: |R⃗| = √(Rₓ² + Rᵧ²)

• Find the resultant direction: θ = arctan(Rᵧ / Rₓ), adjusted for quadrant using the signs of Rₓ and Rᵧ.

4.9 Coordinate Systems

A coordinate system is a mathematical framework that assigns a unique set of numbers to every point in space. It provides a reference structure within which positions, directions, and vector components can be precisely described. Without a coordinate system, statements like “the ball is 3 meters to the left” are ambiguous: left relative to what?

In introductory physics, the most commonly used coordinate system is the Cartesian coordinate system, named after René Descartes. In two dimensions, it consists of two perpendicular number lines — the x-axis (horizontal) and y-axis (vertical) — intersecting at the origin (0, 0). Every point in the plane is identified by an ordered pair (x, y), where x is the horizontal distance from the origin and y is the vertical distance.

In three dimensions, a third axis (z, typically pointing out of the page or upward) is added. A point in 3D space is described by an ordered triple (x, y, z). In this book, nearly all problems are one- or two-dimensional.

Table 4.3 Coordinate systems used in introductory physics. Cartesian (2D) is the most frequently used.

4.10 Position on a Number Line

The simplest coordinate system is a number line: a single axis with a chosen zero point (the origin) and a positive direction. Position is then a single signed number. Positive numbers indicate distance in the positive direction from the origin; negative numbers indicate distance in the negative direction.

The choice of origin and positive direction is yours to make. You choose them to make the problem as simple as possible. Common choices: place the origin where the motion starts, or where an interesting event occurs. Point the positive axis in the direction of initial motion, or upward if vertical motion is involved.

Position: x = +3.0 m means 3.0 m in the positive direction from the origin

Position: x = −2.0 m means 2.0 m in the negative direction from the origin

Example — One-dimensional position A car starts at x = 0 and drives 5.0 m to the right: x = +5.0 m. If it then drives 8.0 m to the left: x = +5.0 − 8.0 = −3.0 m. The car is 3.0 m to the left of the origin.

Common Mistake: Position (x = −3.0 m) and distance (3.0 m from origin) are different quantities. Position is a signed number; it can be negative. Distance is a magnitude; it is always positive. Confusing them is one of the most common errors in kinematics.

4.11 Position in Two Dimensions

In two-dimensional problems, position requires two numbers: one for each axis. The position vector ⃗r of a point specifies its location relative to the origin. Its components are the x and y coordinates of the point.

⃗r = (x, y) or equivalently ⃗r = xî + yĵ

Here î is the unit vector pointing in the +x direction and ĵ is the unit vector pointing in the +y direction. Unit vectors have magnitude 1 and indicate direction only.

The magnitude of the position vector is the straight-line distance from the origin to the point:

|⃗r| = √(x² + y²)

The direction of the position vector is the angle θ = arctan(y/x), measured counterclockwise from the positive x-axis.

Example — Position vector A bird is located 30 m east and 40 m north of a reference tree. Its position vector is ⃗r = (30 m, 40 m). Its distance from the tree is √(30² + 40²) = 50 m. Its direction is arctan(40/30) = 53° north of east.

When a particle moves, its position vector changes. The change in position vector from initial position ⃗rᵢ to final position ⃗rᴍ is the displacement vector ∆⃗r = ⃗rᴍ − ⃗rᵢ. Displacement, velocity, acceleration, and force are all vectors described this way.

Bridge Note In the advanced series, position vectors in three dimensions are written r = xî + yĵ + zᵏ and manipulated using the full machinery of linear algebra and tensor calculus. The two-dimensional notation developed here is the starting point for that treatment.

4.12 Choosing a Useful Coordinate System

One of the key skills in physics problem-solving is choosing a coordinate system that simplifies the algebra. The physics is real and independent of the coordinate system you choose; the mathematics, however, is much easier with a wise choice.

Guidelines for Choosing a Coordinate System

• Align an axis with the direction of motion or the direction of the dominant force. This often makes one component zero, reducing a two-dimensional problem to two one-dimensional ones.

• Place the origin at the starting point of the motion, or at a pivot, or at the point where the most interesting event occurs. This usually makes one or both initial coordinates zero.

• For inclined-plane problems, align one axis with the slope. The component of gravity along the slope is a simple g sinθ; the component perpendicular to the slope is g cosθ. Neither requires a full vector decomposition.

• For projectile motion, align x horizontally and y vertically. Gravity has no x-component; that makes the horizontal motion trivially constant.

• Once you choose a coordinate system, define positive and negative directions clearly in your diagram before writing any equations. Inconsistent sign conventions cause nearly as many errors as wrong formulas.

Example — Inclined plane coordinate choice A box slides down a ramp at 25°. Choosing axes parallel and perpendicular to the ramp makes gravity’s components g sin 25° (along ramp) and g cos 25° (perpendicular). This is simpler than resolving every force twice in a standard x-y system.

Common Mistake: Whatever coordinate system you choose, stick with it throughout the problem. Switching sign conventions mid-calculation is a guaranteed source of sign errors.

4.13 Common Mistakes with Vectors and Coordinates

Mistake 1: Treating a vector quantity as a scalar

Speed and velocity are different. If a ball moves at 10 m/s north for 2 s and then 10 m/s south for 2 s, its average speed is 10 m/s but its average velocity is 0. Treating velocity as just a number (without direction) gives wrong answers in any problem where direction changes.

Mistake 2: Adding vector magnitudes directly

If a person walks 3 km north and 4 km east, the total displacement is not 7 km. It is 5 km (northeast), found by the Pythagorean theorem. Adding magnitudes only works when both vectors point in exactly the same direction.

Mistake 3: Using the wrong angle in component equations

The equations Fₓ = F cosθ and Fᵧ = F sinθ apply when θ is measured from the positive x-axis. If θ is given from another reference (vertical, the slope, north), adjust accordingly. Always draw the diagram and identify the angle explicitly.

Mistake 4: Ignoring the sign of a component

A component is a signed scalar. A force of 30 N pointing left has a negative x-component: Fₓ = −30 N. If you write +30 N, the force points the wrong way in your equations. The sign carries the direction information; dropping it loses that information.

Mistake 5: Confusing position and displacement

Position (x) tells you where an object is relative to the origin. Displacement (Δx) tells you how far and in which direction the position changed. An object at x = −5 m is 5 m to the left of the origin; its displacement from x = +3 m to x = −5 m is Δx = −8 m. These are different numbers with different meanings.

Mistake 6: Not specifying a reference direction

Saying a force is “40 N at 30°” is incomplete. 30° from what? Always state the reference: “30° above the horizontal,” “30° north of east,” or “30° counterclockwise from the positive x-axis.” Without a reference, the angle is ambiguous.

Chapter Summary

Scalars are quantities described by magnitude alone; vectors require both magnitude and direction. The difference is not a mathematical convenience — it reflects the physical reality that many quantities depend fundamentally on direction.

• Scalar quantities: distance, speed, mass, temperature, energy, time, pressure.

• Vector quantities: displacement, velocity, acceleration, force, momentum, electric field, weight.

• Vectors are represented graphically as arrows; the length represents magnitude and the direction of the arrowhead indicates direction.

• Vector notation: boldface (v) in print; overhead arrow (⃗v) in handwriting. Magnitude is written |v⃗| or simply v.

• Vector addition: tip-to-tail method; the resultant runs from the tail of the first vector to the tip of the last.

• Vector subtraction: A⃗ − B⃗ = A⃗ + (−B⃗), where −B⃗ has the same magnitude as B⃗ but opposite direction.

• Components: Aₓ = |A⃗| cosθ and Aᵧ = |A⃗| sinθ, where θ is measured from the positive x-axis. Add vectors by components for precision.

• Coordinate systems assign number pairs (or triples) to points in space. Cartesian coordinates (x, y) are most common in this book.

• Choose a coordinate system that aligns axes with motion or forces, places the origin at a convenient starting point, and defines sign conventions clearly before writing equations.

Key Terms

scalar: A physical quantity described completely by a magnitude (a number and a unit) with no direction.

vector: A physical quantity described by both a magnitude and a direction.

magnitude: The size of a vector quantity; always non-negative. Written |A⃗| or A.

resultant: The single vector that is equivalent to the sum of two or more vectors; found by tip-to-tail addition or the component method.

tip-to-tail method: A graphical technique for adding vectors: the tail of each successive vector is placed at the tip of the previous one; the resultant runs from the first tail to the last tip.

component: The projection of a vector onto a coordinate axis; a signed scalar. For vector A⃗ at angle θ from the x-axis: Aₓ = A cosθ, Aᵧ = A sinθ.

resolving a vector: Decomposing a vector into its components along perpendicular coordinate axes.

unit vector: A dimensionless vector of magnitude exactly 1, used to indicate direction. î points along +x, ĵ points along +y.

coordinate system: A mathematical framework that assigns unique numerical coordinates to every point in space; provides the reference structure for describing position and direction.

Cartesian coordinate system: A coordinate system using perpendicular axes (x and y in 2D; x, y, z in 3D) intersecting at the origin.

origin: The reference point in a coordinate system, assigned coordinates (0, 0) in 2D or (0, 0, 0) in 3D.

position vector: A vector from the origin to a point in space; its components are the coordinates of that point.

displacement: The vector change in position: Δ⃗r = ⃗r_f − ⃗r_i; has both magnitude (distance traveled in a straight line) and direction.

positive direction: The arbitrarily chosen direction along an axis in which the coordinate value increases; opposite to the negative direction.

Concept Review Questions

Answer in complete sentences. No calculation required.

1. What is the difference between a scalar and a vector? Give two examples of each from everyday experience.

2. Why is speed a scalar but velocity a vector? Give a physical example where treating velocity as a scalar would give the wrong answer.

3. An arrow representing a vector is moved 10 cm to the right without rotating or stretching it. Does the vector change? Explain.

4. Can the resultant of two vectors ever be zero? If yes, describe the conditions. If no, explain why not.

5. Explain in your own words what “resolving a vector into components” means. Why is this useful?

6. Why is the magnitude of A⃗ + B⃗ not generally equal to |A⃗| + |B⃗|?

7. What is a coordinate system and why do physicists use them?

8. Why is the choice of coordinate system not physically significant, even though it affects the appearance of the equations?

9. A car travels 60 km/h. Is this a scalar or a vector? A car travels 60 km/h northbound on Interstate 5. Is this a scalar or a vector?

10. Explain the difference between position and displacement in one-dimensional motion. Can displacement be negative? Can position be negative?

Vector Drawing Exercises

For each problem, draw an accurate arrow diagram on graph paper. Label all vectors with symbol, magnitude, and unit. Include a compass rose or axis labels.

11. Draw a vector representing a force of 40 N to the right. Then draw a second vector representing a force of 25 N upward. Use the tip-to-tail method to find the resultant. Estimate the magnitude and direction of the resultant from your diagram.

12. Draw a velocity vector of 30 m/s at 45° above the horizontal. Graphically resolve it into horizontal and vertical components. Label each component with its approximate value (you may use trigonometry to check).

13. A boat travels 6.0 km east, then 8.0 km north. Draw both displacement vectors tip-to-tail. Draw the resultant. Measure its length and direction from your diagram. Verify with the Pythagorean theorem.

14. Draw a force vector of 50 N at 120° counterclockwise from the positive x-axis. Find its x and y components graphically, then verify with Fₓ = 50 cos 120° and Fᵧ = 50 sin 120°.

15. Two forces act on an object: 20 N east and 20 N west. Draw both, apply tip-to-tail addition, and find the resultant. What does this result mean physically?

Component Practice Problems

Show all work. Resolve each vector into components using Aₓ = A cosθ and Aᵧ = A sinθ. Give answers to three significant figures.

16. A displacement vector has magnitude 12.0 m and points at 35.0° above the positive x-axis. Find its x and y components.

17. A force of 75.0 N acts at 60.0° above the horizontal. Find Fₓ and Fᵧ.

18. A velocity of 20.0 m/s acts at 15.0° below the horizontal (i.e., −15.0° from the x-axis). Find vₓ and vᵧ.

19. Three forces act on an object: F⃗₁ = 30.0 N at 0°; F⃗₂ = 40.0 N at 90°; F⃗₃ = 20.0 N at 180°. Find the magnitude and direction of the resultant.

20. A hiker walks 5.00 km at 30.0° north of east, then 3.00 km due north. Find the magnitude and direction of the total displacement.

21. A vector has components Aₓ = −4.0 m and Aᵧ = +3.0 m. Find its magnitude and the angle it makes with the positive x-axis (be careful about quadrant).

22. Two vectors: A⃗ = 10.0 m at 0° and B⃗ = 10.0 m at 60°. Find A⃗ + B⃗ (magnitude and direction).

23. Repeat the previous problem to find A⃗ − B⃗ (magnitude and direction).

Coordinate System Practice

Use a standard Cartesian coordinate system (x rightward positive, y upward positive) unless otherwise stated.

24. Plot the following points: A = (3, 4), B = (−2, 5), C = (0, −3), D = (−4, −1). For each, find the distance from the origin.

25. A particle is at position (6.0 m, 8.0 m). A second particle is at (1.0 m, 2.0 m). Find the displacement vector from the second particle to the first, and its magnitude.

26. A car starts at x = +20.0 m. It drives to x = −15.0 m. (a) What is the displacement? (b) What is the distance traveled?

27. A ball is thrown horizontally from a cliff. Set up a coordinate system with origin at the launch point, x horizontal, y vertical. What are the initial position coordinates? What is the sign of y after the ball falls?

28. A box sits on a ramp angled at 20° to the horizontal. Set up a coordinate system aligned with the ramp (x’ along the ramp, y’ perpendicular). What is the component of gravity (g = 9.81 m/s² downward) along the ramp axis? Perpendicular to it?

29. Three cities: City A is at the origin. City B is 100 km due east. City C is 60 km north of city B. A traveler goes from A to C via B. (a) What is the distance traveled? (b) What is the displacement from A to C (magnitude and direction)?

Applied Physics Problems

Apply component methods. Draw a diagram for each problem. Give final answers in SI units with correct significant figures.

30. A soccer player kicks a ball with a force of 120 N at 20.0° above the ground. (a) What is the horizontal component of the force? (b) The vertical component?

31. An airplane flies with a velocity of 250 m/s at 15.0° north of due east. (a) What is the eastward component of velocity? (b) The northward component?

32. A rope attached to a sled makes an angle of 40.0° with the ground. The tension in the rope is 200 N. (a) How much force is pulling the sled horizontally? (b) How much force is lifting the sled?

33. A river flows 2.0 m/s due south. A boat points due east and motors at 5.0 m/s relative to the water. (a) What is the velocity of the boat relative to the ground (magnitude and direction)? (b) If the river is 200 m wide, how far downstream does the boat land?

34. Three forces act on a ring: F⃗₁ = 50.0 N east; F⃗₂ = 70.0 N at 120° from east (second quadrant); F⃗₃ = 40.0 N south. Find the resultant force (magnitude and direction).

35. A weight hangs from two ropes attached to the ceiling. Rope A makes 30° with the vertical and has tension T_A. Rope B is vertical and has tension T_B = 60.0 N. If the system is in equilibrium (net force = zero), find T_A and the vertical component of T_A.

Chapter 4 Checkpoint

Before moving on to Chapter 5, confirm you can do each of the following without notes.

Continue to Chapter 5: Graphs, Models, and Experimental Uncertainty →