Compute a Nonnegative Vector Length

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Find the magnitude of vector ⟨6,8⟩. Candidate answers: (A) 10 (B) 11 (C) 20 (D) 9.
• Independent check inputs: Find the magnitude of vector ⟨9,12⟩. Candidate answers: (A) 14 (B) 15 (C) 16 (D) 30.

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. A vector’s component displacements form perpendicular legs of a right triangle. The square root of the sum of squared components gives its magnitude independently of the component signs.

Here is the method on the supplied teaching case. Magnitude is √(horizontal²+vertical²)=√(36+64)=√100=10. It is a nonnegative length, unlike a signed component.

Magnitude is a scalar length, not an ordered pair. Adding component values would fail for opposite signs; squaring and summing preserves their perpendicular contributions to the nonnegative distance.

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.

Check your reasoning

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Take it further

Changed-input task: Find the magnitude of vector ⟨27,36⟩. Candidate answers: (A) 44 (B) 45 (C) 46 (D) 90. Show the mathematical reason, not only the selected answer. Expected result: 45. Magnitude is √(horizontal²+vertical²)=√(729+1296)=√2025=45. It is a nonnegative length, unlike a signed component.

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