§ 20: 20.20

20.20 Знайдіть кут між векторами m = a + b і n = a – 2b, якщо

|a| = 2, |b| = 2, ∠(a, b) = 135°

m = a + b; n = a – 2b; |a| = 2, |b| = 2, ∠(a, b) = 135°.

∠(m, n) – ?

cos∠(m, n) = m ∵ n m|n|

m • n = (a + b) • (a – 2b) = a² – 2ab + ab – 2b² = 2 – 2 • 2 •

• (–22) – 2 • 4 = 2 + 2 – 8 = –4.

|m| = a+ b² = a2+ 2ab+ b2 = 2+2 • 2 •2 •-22+ 4 =

= 2-4+4 = 2.

|n| = a- 2b)2 = 2-4 • 2•2 •-22+ 4 •4 = 2+8+16 =

= 26.

cos ∠(m, n) = -42• 26 = –4213 = –213 = –21313

∠(m, n) = 180° – arccos 21313.