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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 the vectors <3, 2> and <2, 3>.

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

Answer: <5, 5>

Practice problems

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

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

Answer: <5, 5>

2. Subtract the vectors: <4, 3> - <3, 4>.

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

Answer: <1, -1>

3. Find the scalar multiple 2<5, 1>.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the scalar multiple 2<5, 1>.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Multiply each component by 2.
  4. 2 times 5 = 10, and 2 times 1 = 2.
  5. So 2<5, 1> = <10, 2>.
  6. Check the result by substituting or estimating: the response should match <10, 2> and make sense in the original problem.

Answer: <10, 2>

4. Find the magnitude of the vector <9, 12>.

Show solution
  1. The magnitude is sqrt(x^2 + y^2).
  2. sqrt(9^2 + 12^2) = sqrt(81 + 144) = sqrt(225).
  3. sqrt(225) = 15, so the magnitude is 15.

Answer: 15

5. Find the dot product <3, 3> . <1, 3>.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the dot product <3, 3> . <1, 3>.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Multiply matching components, then add.
  4. (3)(1) + (3)(3) = 3 + 9.
  5. The dot product is 12.
  6. Check the result by substituting or estimating: the response should match 12 and make sense in the original problem.

Answer: 12

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