(a) Three vectors a, b and c are (begin{pmatrix} 8 \ 3 end{pmatrix}, begin{pmatrix} 6 \ -5 end{pmatrix}) and (begin{pmatrix} 2 \ -3 end{pmatrix}) respectively. Find the vector d such that (|d| = sqrt{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 AOB = 30°, find, correct to 2 decimal places, the values of n.
Explanation
(a) (a = begin{pmatrix} 8 \ 3 end{pmatrix} ; b = begin{pmatrix} 6 \ -5 end{pmatrix} ; c = begin{pmatrix} 2 \ -3 end{pmatrix} ; |d| = sqrt{41})
(p = a + b – 2c)
= (begin{pmatrix} 8 \ 3 end{pmatrix} + begin{pmatrix} 6 \ -5 end{pmatrix} + 2begin{pmatrix} 2 \ -3 end{pmatrix})
= (begin{pmatrix} 8 + 6 + 4 \ 3 – 5 – 6 end{pmatrix})
= (begin{pmatrix} 10 \ -8 end{pmatrix})
(|p| = sqrt{10^{2} + (-8)^{2}} = sqrt{164})
(|p| = 2sqrt{41})
(|d| = frac{1}{2} |p|)
(therefore d = frac{1}{2} p = frac{1}{2} begin{pmatrix} 10 \ -8 end{pmatrix})
= (begin{pmatrix} 5 \ -4 end{pmatrix})
(b) A(3, 4) ; B(3, n).
(OA = 3i + 4j ; OB = 3i + nj)
(OA cdot OB = |OA||OB| cos theta)
((3i + 4j) cdot (3i + nj) = |3i + 4j||3i + nj| cos 30)
(9 + 4n = 5 (sqrt{9 + n^{2}}) (sqrt{3}{2}))
(2(9 + 4n) = 5(sqrt{3(9 + n^{2})}))
(18 + 8n = 5sqrt{27 + 3n^{2}})
Squaring both sides,
(324 + 288n + 64n^{2} = 25(27 + 3n^{2}))
(324 + 288n + 64n^{2} = 675 + 75n^{2})
(75n^{2} – 64n^{2} – 288n + 675 – 324 = 0)
(11n^{2} – 288n + 351 = 0)
(n = frac{-(-288) pm sqrt{(-288)^{2} – 4(11)(351)}}{2(11)})
(n = frac{288 pm sqrt{82944 – 15444}}{22})
(n = frac{288 pm sqrt{67500}}{22})
(n = frac{288 pm 259.81}{22})
(n = frac{288 + 259.81}{22} ; n = frac{288 – 259.81}{22})
(n = frac{547.81}{22} ; n = frac{28.19}{22})
(n = 24.9 ; n = 1.28)