Add and subtract vectors
You will be able to: Add and subtract vectors component by component and interpret the resultant.
Why can two nonzero vectors add to zero?
A robot first moves 3 m east and 2 m north, then 1 m west and 3 m north. To find where it finishes, keep separate totals for east–west and north–south motion.
A useful starting point: Vectors, components and unit vectors →
Words and symbols before equations
- Resultant
- The vector sum: one vector with the same combined effect.
- Head-to-tail
- Place the start of the second arrow at the end of the first without rotating it.
- Vector subtraction
- A − B means A + (−B): reverse every component of B.
- Δ, delta
- A change: final value minus initial value.
What this picture assumes
Two consecutive displacements. A = (3, 2) m is fixed. Read the labeled coordinates; horizontal and vertical screen scales differ. The resultant joins the start to the final endpoint.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- A = (3, 2) m; B = (-1, 3) m; sum = (2, 5) m, magnitude 5.385 m.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
For A = (3, 2) m and B = (−1, 3) m, the resultant is (2, 5) m. Add x to x and y to y. The head-to-tail diagram and component arithmetic describe the same endpoint.
Its magnitude is √29 m, about 5.39 m, not |A| + |B|. Magnitudes add directly only when the arrows point in the same direction. Opposite equal arrows cancel even though each has nonzero magnitude.
Subtracting vectors is essential for a change in position or velocity. If v changes from (3, 0) m/s to (0, 3) m/s, Δv = (−3, 3) m/s. The speed remains 3 m/s, but velocity changes because its direction changes.
A worked example, step by step
Let A = (4, −2) m and B = (−1, 5) m. Find A + B and A − B.
- Write each pair in the same coordinate frame.
- A + B = (4 − 1, −2 + 5) m = (3, 3) m.
- Reverse B to subtract: −B = (1, −5) m, so A − B = (5, −7) m.
- The sum points northeast. The difference points southeast; neither is found by subtracting magnitudes.
Subtract the final and initial vectors in the stated order; a difference of magnitudes does not give the change in a vector.
An object turns at constant speed. Is Δv zero?
Compare with an explanation
No, unless the direction returns to its starting direction over the chosen interval. Velocity includes direction.
Predict. Change one thing. Explain.
Keep A fixed and change B until the resultant vanishes. Explain why each component must vanish separately.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
A = (3, 2) m; B = (-1, 3) m; sum = (2, 5) m, magnitude 5.385 m.
Two consecutive displacements. A = (3, 2) m is fixed. Read the labeled coordinates; horizontal and vertical screen scales differ. The resultant joins the start to the final endpoint.
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 questionA student walks A = (6, 2) m, then B = (−2, 1) m. Find the resultant, its magnitude, and the vector that would return the student directly to the start.
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Compare with the answer and four-point rubric
- 1 point: Add components to obtain R = (4, 3) m.
- 1 point: Magnitude √(4² + 3²) = 5 m.
- 1 point: The direct return is −R = (−4, −3) m.
- 1 point: The return cancels both components and has magnitude 5 m, rather than the total distance already walked.
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 1How do vectors add algebraically?
Add corresponding components in a common coordinate system.
RECALL 2What does −A do?
Reverses direction; it preserves magnitude.
RECALL 3Can unchanged speed accompany changing velocity?
Yes. Changing direction changes velocity.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Add and subtract vectors
- (A + B)ₓ = Aₓ + Bₓ; (A + B)ᵧ = Aᵧ + Bᵧ.
- A − B = A + (−B).
- Δv = v_final − v_initial, including direction.
Remember: Subtract the final and initial vectors in the stated order; a difference of magnitudes does not give the change in a vector.
Conditions: Two consecutive displacements. A = (3, 2) m is fixed. Read the labeled coordinates; horizontal and vertical screen scales differ. The resultant joins the start to the final endpoint.
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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