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Grade 12 · lesson 1 of 4

Combine Directed Displacements by Component

Adding vectors combines horizontal components and vertical components separately.

About 25 minutes, plus any extended practice

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Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: Add vectors ⟨2,−3⟩ and ⟨3,2⟩. Candidate answers: (A) ⟨-1,-5⟩ (B) ⟨4,0⟩ (C) ⟨5,-1⟩ (D) ⟨6,-6⟩.
  • Independent check inputs: Add vectors ⟨3,−4⟩ and ⟨3,2⟩. Candidate answers: (A) ⟨0,-6⟩ (B) ⟨5,-1⟩ (C) ⟨6,-2⟩ (D) ⟨9,-8⟩.

Learn the idea

Vectors represent directed quantities through components, while parametric equations describe coordinates using a shared parameter. Component arithmetic, magnitude and dot product answer different questions and therefore produce different kinds of results. Adding vectors combines horizontal components and vertical components separately. The resulting displacement can be drawn by following one vector after the other, with the sum connecting starting and ending locations.

Here is the method on the supplied teaching case. Add matching components: horizontal 2+3=5, vertical −3+2=-1. This represents combining directed displacements.

A vector sum keeps two components; it is not a sum of magnitudes. Opposing directions use negative components, so preserve signs and match horizontal with horizontal rather than crossing component positions.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: Add vectors ⟨2,−3⟩ and ⟨3,2⟩. Candidate answers: (A) ⟨-1,-5⟩ (B) ⟨4,0⟩ (C) ⟨5,-1⟩ (D) ⟨6,-6⟩. Solution: Add matching components: horizontal 2+3=5, vertical −3+2=-1. This represents combining directed displacements. The final answer is ⟨5,-1⟩. A vector sum keeps two components; it is not a sum of magnitudes. Opposing directions use negative components, so preserve signs and match horizontal with horizontal rather than crossing component positions.

Your turn

Check your reasoning

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Helpful tips
  • Adding vectors combines horizontal components and vertical components separately.
  • A vector sum keeps two components; it is not a sum of magnitudes.
Add vectors ⟨3,−4⟩ and ⟨3,2⟩.

Feedback appears here after you check. This is practice, not a graded school assessment.

See the answer and explanation

⟨6,-2⟩

Add matching components: horizontal 3+3=6, vertical −4+2=-2. This represents combining directed displacements.

Take it further

Changed-input task: Add vectors ⟨9,−10⟩ and ⟨3,2⟩. Candidate answers: (A) ⟨11,-7⟩ (B) ⟨12,-8⟩ (C) ⟨27,-20⟩ (D) ⟨6,-12⟩. Show the mathematical reason, not only the selected answer. Expected result: ⟨12,-8⟩. Add matching components: horizontal 9+3=12, vertical −10+2=-8. This represents combining directed displacements.

Nothing here is saved to an account or sent to a teacher. The answer download includes your current choice and any checked feedback. It does not include choices from the separate More practice section. Download that section separately.

More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. Add vectors ⟨4,−5⟩ and ⟨3,2⟩.
Need a hint?
  • Adding vectors combines horizontal components and vertical components separately.
  • A vector sum keeps two components; it is not a sum of magnitudes.
2. Add vectors ⟨5,−6⟩ and ⟨3,2⟩.
Need a hint?
  • Adding vectors combines horizontal components and vertical components separately.
  • A vector sum keeps two components; it is not a sum of magnitudes.
3. Add vectors ⟨6,−7⟩ and ⟨3,2⟩.
Need a hint?
  • Adding vectors combines horizontal components and vertical components separately.
  • A vector sum keeps two components; it is not a sum of magnitudes.

Your choices are temporary. Feedback appears after you check all the answers.

Nothing here is saved to an account or sent to a teacher. Keep your own notes or download them before leaving. The score describes these questions; it does not certify subject mastery.

Check the subject sources

These LifeStarter passages, scenarios and exercises are authored for this Academy with AI assistance and review. External resources keep their publishers’ terms. A grade label suggests a starting level; it does not establish alignment with your school’s complete requirements.