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Let i-hat and j-hat be unit vectors perpendicular to each other. Given: p = 3*i + 4*j, q = 5*i, 4*r = p + q, and 2*s = p - q. Which of the following statements is/are true?

1. |r + k*s| = |r - k*s| for all real k
2. r is perpendicular to s
3. r + s is perpendicular to r - s
4. |r| = |s| = |p| = |q|

Correct answer: r is perpendicular to s

Solution

From the given: r = (p+q)/4 = (8i+4j)/4 = 2i+j and s = (p-q)/2 = (-2i+4j)/2 = -i+2j. Then r.s = (2)(-1)+(1)(2) = 0, confirming r is perpendicular to s. Since r perp s, |r+ks|² = |r|² + k²|s|² = |r-ks|² for all k. Also (r+s).(r-s) = |r|² - |s|² = 5 - 5 = 0, so r+s perp r-s. However |r|=|s|=sqrt(5) != |p|=5 or |q|=5, so D is false. Statements A, B, C are all true.

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