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Vectors in the Plane

A free Precalculus lesson from the “Parametric, Polar, Vectors, and Intro to Limits” unit, with a worked example and practice problems including step-by-step solutions.

A vector has size and direction. Components let you add vectors one coordinate at a time. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.

What you'll learn

Why it matters: Parametric, polar, vector, and limit ideas prepare students for motion, curves, and the rate-of-change thinking used in Calculus.

Worked example

Problem. Add vectors <2, 3> and <-1, 2>.

  1. Worked Example: First identify exactly what the question is asking: Add vectors <2, 3> and <-1, 2>.
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. Add x-components together.
  4. Add y-components together.
  5. The result is <1, 5>.
  6. Check the result by substituting or estimating: the response should match <1, 5> and make sense in the original problem.

Answer: <1, 5>

Practice problems

1. Add vectors <2, 3> and <-1, 2>.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Add vectors <2, 3> and <-1, 2>.
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. Add x-components together.
  4. Add y-components together.
  5. The result is <1, 5>.
  6. Check the result by substituting or estimating: the response should match <1, 5> and make sense in the original problem.

Answer: <1, 5>

2. Add vectors <3, 4> and <0, 3>.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Add vectors <3, 4> and <0, 3>.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Add x-components together.
  4. Add y-components together.
  5. The result is <3, 7>.
  6. Check the result by substituting or estimating: the response should match <3, 7> and make sense in the original problem.

Answer: <3, 7>

3. Add vectors <4, 2> and <1, 4>.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Add vectors <4, 2> and <1, 4>.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Add x-components together.
  4. Add y-components together.
  5. The result is <5, 6>.
  6. Check the result by substituting or estimating: the response should match <5, 6> and make sense in the original problem.

Answer: <5, 6>

4. Find the magnitude of vector <3, 4>.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the magnitude of vector <3, 4>.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Use magnitude sqrt(a^2 + b^2).
  4. sqrt(3^2 + 4^2) = sqrt(25).
  5. The magnitude is 5.
  6. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

5. Find the magnitude of vector <3, 4>. (variation 2)

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the magnitude of vector <3, 4>.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Use magnitude sqrt(a^2 + b^2).
  4. sqrt(3^2 + 4^2) = sqrt(25).
  5. The magnitude is 5.
  6. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

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