In the coordinate plane, vector vec{p} is perpendicular to vector vec{q}. The magnitude of vec{p} is 10sqrt(34) and the magnitude...

1. TRANSLATE the problem information

• Given information:
  • Vector \(\mathrm{p} \perp \mathrm{q}\) (perpendicular)
  • \(|\mathrm{p}| = 10\sqrt{34}\)
  • \(|\mathrm{q}| = 24\sqrt{34}\)
  • Find: \(|\mathrm{p} + \mathrm{q}|\)

2. INFER the mathematical approach

• Since the vectors are perpendicular, they form a right angle
• This means we can use the Pythagorean theorem: \(|\mathrm{p} + \mathrm{q}|^2 = |\mathrm{p}|^2 + |\mathrm{q}|^2\)
• The sum vector forms the hypotenuse of a right triangle

3. SIMPLIFY by calculating the squares

• \(|\mathrm{p}|^2 = (10\sqrt{34})^2 = 10^2 \times (\sqrt{34})^2 = 100 \times 34 = 3400\)
• \(|\mathrm{q}|^2 = (24\sqrt{34})^2 = 24^2 \times (\sqrt{34})^2 = 576 \times 34 = 19584\)

4. SIMPLIFY by adding the squares

• \(|\mathrm{p} + \mathrm{q}|^2 = 3400 + 19584 = 22984\)
• Factor out: \(22984 = 676 \times 34\)

5. SIMPLIFY by taking the square root

• \(|\mathrm{p} + \mathrm{q}| = \sqrt{676 \times 34} = \sqrt{676} \times \sqrt{34} = 26\sqrt{34}\)

Answer: (D) \(26\sqrt{34}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that perpendicular vectors allow direct application of the Pythagorean theorem

Students might try to add the magnitudes directly:

\(|\mathrm{p} + \mathrm{q}| = |\mathrm{p}| + |\mathrm{q}| = 10\sqrt{34} + 24\sqrt{34} = 34\sqrt{34}\)

This leads them to select Choice (E) \((34\sqrt{34})\)

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors when working with radical expressions

Students might correctly set up \(|\mathrm{p} + \mathrm{q}|^2 = |\mathrm{p}|^2 + |\mathrm{q}|^2\) but then make calculation errors, such as:

• Incorrectly squaring: \((10\sqrt{34})^2 = 10\sqrt{34}\) instead of \(100 \times 34\)
• Forgetting to take the final square root and selecting a choice like \(676 \times 34\)

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students can connect the geometric relationship (perpendicular vectors) to the appropriate mathematical tool (Pythagorean theorem) and then execute the algebraic manipulations accurately with radical expressions.