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Grade 12 · unit review

Review: Precalculus: Vectors and Parametric Lines

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.

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Bring the unit skills together

Use these 4 new cases to check your reasoning. Choices are temporary and are not saved to an account, sent to staff or recorded as school grades. Download your current review choices before leaving or reloading.

Unit review questions
1. Add vectors ⟨7,−8⟩ 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. Find the magnitude of vector ⟨21,28⟩.
Need a hint?
  • A vector’s component displacements form perpendicular legs of a right triangle.
  • Magnitude is a scalar length, not an ordered pair.
3. Find ⟨7,8⟩·⟨3,−2⟩.
Need a hint?
  • A dot product multiplies corresponding components and sums the products.
  • The result is a number, not a vector of separate products.
4. Eliminate t from x=t+7, y=2t+8, with t any real number.
Need a hint?
  • A parametric line defines x and y using the same parameter.
  • A parameter restriction could limit the resulting graph to a segment or ray.

The download contains your current choices and any checked feedback. Keep the separate transfer-task work in your own document; downloading its instructions does not mark it complete.

Use the skills in a new task

Create a four-part mathematical solution sheet for these supplied cases. 1. Add vectors ⟨9,−10⟩ and ⟨3,2⟩. Candidate answers: (A) ⟨11,-7⟩ (B) ⟨12,-8⟩ (C) ⟨27,-20⟩ (D) ⟨6,-12⟩. 2. Find the magnitude of vector ⟨27,36⟩. Candidate answers: (A) 44 (B) 45 (C) 46 (D) 90. 3. Find ⟨9,10⟩·⟨3,−2⟩. Candidate answers: (A) 14 (B) 6 (C) 7 (D) 8. 4. Eliminate t from x=t+9, y=2t+10, with t any real number. Candidate answers: (A) y=2x+10 (B) y=2x+28 (C) y=2x−8 (D) y=x+1.

Work on paper or in your own document. Explain the choices you made.

  • State each requested quantity or relationship and its input restrictions.
  • Show intermediate calculations or the applicable theorem with matching correspondences.
  • Give all four results, including exact values or requested rounding and units.
  • Check a result by substitution, reversal, dimensions, a representation or a counterexample as appropriate.
Compare with a model response

1. ⟨12,-8⟩. Add matching components: horizontal 9+3=12, vertical −10+2=-8. This represents combining directed displacements. 2. 45. Magnitude is √(horizontal²+vertical²)=√(729+1296)=√2025=45. It is a nonnegative length, unlike a signed component. 3. 7. The dot product multiplies matching components and adds: 9×3+10×(−2)=7. It is a scalar, not a two-component vector. 4. y=2x−8. From x=t+9, t=x−9. Substitute into y: y=2(x−9)+10=2x−8. No parameter restriction removes points here.

A different response can also meet the criteria. Compare the evidence and reasoning, rather than copying the wording.