1) Координаты вектора ¯d в базисе ¯a,¯b,¯c:
¯d = x¯a + y¯b + z¯c

Домножим каждый вектор на координату и сложим:
x = ¯d∙¯a / |¯a|^2 = (4,5,2)∙(5,7,8) / (4^2 + 5^2 + 2^2) = 76 / 45 ≈ 1.69
y = ¯d∙¯b / |¯b|^2 = (4,5,2)∙(3,0,1) / (3^2 + 0^2 + 1^2) = 18 / 10 = 1.8
z = ¯d∙¯c / |¯c|^2 = (4,5,2)∙(-1,4,2) / (-1^2 + 4^2 + 2^2) = 11 / 21 ≈ 0.52

Таким образом, координаты вектора ¯d в базисе ¯a,¯b,¯c равны (1.69, 1.8, 0.52).

2) ¯a∙¯b = (4,5,2)∙(3,0,1) = 43 + 50 + 2*1 = 12 + 0 + 2 = 14

3) ¯c∙¯d = (-1,4,2)∙(5,7,8) = -15 + 47 + 2*8 = -5 + 28 + 16 = 39

4) (2¯a+3¯b)∙(5¯c-4¯d) = (24 + 33)∙(5(-1) - 45) = (8 + 9)∙(-5 - 20) = 17∙(-25) = -425

5) ¯aׯb = (4,5,2)×(3,0,1) = (51 - 20, 23 - 41, 40 - 53) = (5, 6, -15)

6) ¯cׯd = (-1,4,2)×(5,7,8) = (48 - 27, 25 - (-1)8, (-1)7 - 45) = (32 - 14, 10 + 8, -7 - 20) = (18, 18, -27)

7) (¯aׯc)∙¯d = ((4,5,2)×(-1,4,2))∙(5,7,8) = (18, -10, 24)∙(5,7,8) = 185 - 107 + 24*8 = 90 - 70 + 192 = 212