Vectors, components and unit vectors
You will be able to: Resolve a vector into signed components and recover its magnitude and direction.
How can two numbers describe one arrow?
A floor robot moves 3 m east and 4 m north. Its straight displacement is 5 m, but “5 m” alone cannot tell another robot where to go. The direction completes the description.
A useful starting point: Scalars, vectors and direction →
Words and symbols before equations
- Scalar
- A quantity without a spatial direction, such as distance or speed.
- Vector
- A quantity with magnitude and direction, such as displacement.
- Component
- The signed amount of a vector along a chosen coordinate axis.
- î, ĵ, k̂
- Dimensionless unit vectors, each of magnitude one, pointing along positive x, y and z.
- Magnitude |A|
- Nonnegative length of a vector; it retains the vector’s physical unit.
What this picture assumes
Fixed physical axes; camera rotation changes only the projection. Flat view shows x and y; any z component is still reported. The spatial view projects the same vector onto the screen. Each positive axis tip marks +6 m.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- Components (3, 4, 0) m; magnitude 5 m. Divide each component by the magnitude for a unit direction.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
Choose +x east and +y north. Write the displacement as r = (3 m)î + (4 m)ĵ. These are two perpendicular contributions to one vector, not two additional displacements to add to its magnitude.
The right triangle gives |r| = √(3² + 4²) m = 5 m. If θ is measured counterclockwise from +x, Aₓ = A cos θ and Aᵧ = A sin θ. Components may be negative even when the magnitude is positive. Check the quadrant before interpreting an inverse tangent.
The unit direction r̂ = r/|r| = 0.6î + 0.8ĵ carries direction without metres. Dividing the zero vector by its magnitude is undefined. A third component adds k̂; rotate the optional view to see why camera angle changes the projection but not the vector. Calculations in this motion unit remain in one or two dimensions.
A worked example, step by step
A displacement has components −3 m east and +4 m north. Find its magnitude and unit direction.
- The x component is −3 m, meaning west; y is +4 m, meaning north.
- Use perpendicular components: |r| = √[(−3)² + 4²] m.
- The magnitude is 5 m; the direction is northwest, 126.9° counterclockwise from +x.
- Divide each component by 5 m: r̂ = −0.6î + 0.8ĵ. Its magnitude is one and it has no unit.
A negative component describes direction. It does not make the vector’s magnitude negative.
Does rotating the camera make a 5 m vector shorter?
Compare with an explanation
No. Its screen projection can shorten, but its physical magnitude stays 5 m.
Predict. Change one thing. Explain.
Keep x = 3 m and y = 4 m. Set View to 1 (spatial), rotate the camera, then return to View 0 (flat). Predict whether the numerical components change. Next change only x to −3 m.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Components (3, 4, 0) m; magnitude 5 m. Divide each component by the magnitude for a unit direction.
Fixed physical axes; camera rotation changes only the projection. Flat view shows x and y; any z component is still reported. The spatial view projects the same vector onto the screen. Each positive axis tip marks +6 m.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant position, velocity, acceleration or integral relationship to justify your prediction.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionFor r = (8 m)î + (6 m)ĵ, state the components, calculate the magnitude, give r̂, and explain whether a camera rotation changes any of these physical quantities.
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Compare with the answer and four-point rubric
- 1 point: Components are +8 m in x and +6 m in y.
- 1 point: Magnitude is √(64 + 36) = 10 m.
- 1 point: r̂ = 0.8î + 0.6ĵ, dimensionless.
- 1 point: A camera rotation changes the screen projection only; the fixed coordinate components and magnitude stay the same.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What does î mean?
A unit vector in the +x direction, with magnitude one and no physical unit.
RECALL 2Can a component be negative?
Yes; the sign identifies its direction relative to the chosen axis.
RECALL 3How do you form a unit direction?
Divide a nonzero vector by its magnitude.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Vectors, components and unit vectors
- A = Aₓî + Aᵧĵ (+ A_zk̂ for spatial notation).
- In 2D: |A| = √(Aₓ² + Aᵧ²). In 3D add A_z² inside the square root. Â = A/|A| for nonzero A.
- Aₓ = A cos θ; Aᵧ = A sin θ when θ is measured from +x.
Remember: A negative component describes direction. It does not make the vector’s magnitude negative.
Conditions: Fixed physical axes; camera rotation changes only the projection. Flat view shows x and y; any z component is still reported. The spatial view projects the same vector onto the screen. Each positive axis tip marks +6 m.
Refresh Kid · AP Physics C: Mechanics Unit 1 (official Unit 1) · Objectives 1.1.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 1.1, objectives 1.1.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. This is Mechanics Unit 1: Kinematics. Quantitative motion problems stay in one or two dimensions; the optional spatial view clarifies vector notation. Students should study calculus alongside this course. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
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