Vector components, magnitude and combinations
Represent directed quantities with components and distinguish magnitude from direction.
What you will tackle
- Add/subtract vectors and multiply by scalars
- Find magnitude from components
- Interpret component operations in a contextual model
Operate component by component
For u = ⟨a,b⟩ and v = ⟨c,d⟩, u + v = ⟨a + c,b + d⟩ and ku = ⟨ka,kb⟩. A negative scalar reverses direction. The magnitude √(a² + b²) is nonnegative and follows from the right triangle formed by perpendicular components.
Interpret the coordinate directions
A displacement of ⟨−3,4⟩ means three units in the negative horizontal direction and four in the positive vertical direction. Its length is 5, not the sum 1 or 7. The vector between positions P and Q is Q − P; reversing the order reverses direction.
Build your notes, then practise
Work on paper before opening an explanation. Guided tasks provide a starting point; independent tasks remove the scaffolding; application tasks ask you to justify a method or interpretation in context.
Guided practice
Guided 1
Let u = ⟨3,−2⟩ and v = ⟨−1,5⟩. Find u + v, 2u − v and |u|.
Show a starting hint
Keep horizontal and vertical components in their own positions.
Compare with the explained answer
- u + v = ⟨2,3⟩.
- 2u − v = ⟨6,−4⟩ − ⟨−1,5⟩ = ⟨7,−9⟩.
- |u| = √(3² + (−2)²) = √13.
Answer / response criteria: ⟨2,3⟩; ⟨7,−9⟩; √13.
Independent practice
Independent 1
P = (−2,1) and Q = (4,9). Find vector PQ, vector QP and their magnitudes.
Compare with the explained answer
- Q − P = ⟨6,8⟩.
- P − Q = ⟨−6,−8⟩.
- Both magnitudes are √(36 + 64) = 10.
Answer / response criteria: PQ = ⟨6,8⟩; QP = ⟨−6,−8⟩; both magnitudes 10.
Application practice
Application 1
A boat's velocity relative to water is ⟨6,0⟩ km/h and the current is ⟨0,2⟩ km/h. Find its velocity over the ground and its displacement after 1.5 hours, assuming both remain constant.
Compare with the explained answer
- Add velocities to obtain ⟨6,2⟩ km/h.
- Multiply by 1.5 hours: displacement ⟨9,3⟩ km.
- Ground speed is √40 km/h; a straight-line constant-velocity model is assumed.
Answer / response criteria: Ground velocity ⟨6,2⟩ km/h; displacement ⟨9,3⟩ km.
What can you explain now?
A boat's velocity relative to water is ⟨6,0⟩ km/h and the current is ⟨0,2⟩ km/h. Find its velocity over the ground and its displacement after 1.5 hours, assuming both remain constant.
Compare with the explanation
Ground velocity ⟨6,2⟩ km/h; displacement ⟨9,3⟩ km.
Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.
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Source and review status
Coverage follows the named requirements and examples in the 2026–2027 ACT test guide and subject descriptions. That guide explicitly does not list every possible topic. The broader coded College and Career Readiness Standards are also mapped to conceptual teaching and practice; this does not calibrate a learner to a score band. The separate scientific-background context list and reading-complexity rubric have their own stated limits.
Original teaching and explained self-check practice are available. Specialist review and difficulty calibration remain pending. Essays require self-check or teacher review and are not automatically marked or awarded official scores.
Official source specification / guide