A vector in the xy-plane has components langle 8m, 6mrangle where m gt 0. Which equation represents the square of...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
A vector in the xy-plane has components \(\langle 8\mathrm{m}, 6\mathrm{m}\rangle\) where \(\mathrm{m} \gt 0\). Which equation represents the square of the magnitude of this vector?
- \(10\mathrm{m}^2\)
- \(64\mathrm{m}^2 + 36\mathrm{m}^2\)
- \(100\mathrm{m}\)
- \(100\mathrm{m}^2\)
1. TRANSLATE the problem information
- Given information:
- Vector components: \(\langle 8m, 6m \rangle\) where \(m \gt 0\)
- Need to find: equation representing the square of the magnitude
- What this tells us: We need to apply the vector magnitude formula, then square the result
2. INFER the approach
- To find the square of the magnitude, we first find the magnitude using \(|v| = \sqrt{a^2 + b^2}\)
- Then we square that result
- For vector \(\langle 8m, 6m \rangle\), we have \(a = 8m\) and \(b = 6m\)
3. SIMPLIFY the magnitude calculation
- Set up the formula: \(|v| = \sqrt{(8m)^2 + (6m)^2}\)
- Calculate the squares: \(|v| = \sqrt{64m^2 + 36m^2}\)
- Add under the radical: \(|v| = \sqrt{100m^2}\)
- Simplify the square root: \(|v| = 10m\) (since \(m \gt 0\))
4. SIMPLIFY to find the square of the magnitude
- Square the magnitude: \(|v|^2 = (10m)^2\)
- Final result: \(|v|^2 = 100m^2\)
Answer: D (100m²)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students stop at the unsimplified form \(\sqrt{64m^2 + 36m^2}\) and think this represents the square of the magnitude.
They see that \(64m^2 + 36m^2\) appears in choice (B) and select it without recognizing they need to simplify further and then square the result. This leads them to select Choice B (64m² + 36m²).
Second Most Common Error:
Poor TRANSLATE reasoning: Students correctly find the magnitude as \(10m\) but forget the question asks for the square of the magnitude.
They stop at \(|v| = 10m\) and look for an answer choice that matches, potentially getting confused since \(10m\) isn't listed, or they might incorrectly think it should be \(10m^2\) rather than \(100m^2\). This leads to confusion and guessing between choices (A) and (C).
The Bottom Line:
This problem tests whether students can execute a complete two-step process: finding a vector's magnitude and then squaring it. Success requires careful attention to what the question is actually asking for and following through with all algebraic simplification steps.