Theory question 116
Vectors(a) Three vectors a, b and c are (8, 3), (6, −5) and (2, 3) (column vectors) respectively. Find the vector d such that |d| = √41 and d is in the direction of a + b − 2c.
(b) The coordinates of A and B are (3, 4) and (3, n) respectively. If O is the origin and ∠AOB = 30°, find, correct to 2 decimal places, the values of n.
Show the model answer
Model answer
(a) a + b − 2c = (10, −8), magnitude √164 = 2√41.
d = √41 × (10, −8)/(2√41) = (5, −4).
(b) OA·OB = 9 + 4n = |OA||OB| cos 30° = 5√(9 + n²)(√3/2).
Square: 4(9 + 4n)² = 75(9 + n²) gives 11n² − 288n + 351 = 0.
n = (288 ± √67500)/22: n = 24.90 or n = 1.28.
Explanation
Use the dot product for the angle between two vectors.