Theory question 107
VectorsGiven that m = 2i + pj, n = −3i + 4j and cos θ = 6/(5√13), where θ is an angle between m and n. Find the values of p.
Show the model answer
Model answer
m·n = −6 + 4p; |m| = √(4 + p²), |n| = 5.
(4p − 6)/(5√(4 + p²)) = 6/(5√13), so √13(4p − 6) = 6√(4 + p²).
Squaring: 13(16p² − 48p + 36) = 36(4 + p²), giving 43p² − 156p + 81 = 0.
(43p − 27)(p − 3) = 0, so p = 3 or p = 27/43.
Check: cos θ is positive, so 4p − 6 > 0; only p = 3 satisfies the original equation (p = 27/43 gives a negative cosine, an extra root from squaring).
Explanation
Use cos θ = m·n/(|m||n|), and check roots after squaring.