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denysyukd705
@denysyukd705
July 2023
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ДАЮ 100 БАЛІВ!! / ДАЮ 100 БАЛЛОВ!!!
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sghhajjhnz
2.
1)Вектор MN:
MN = N - M = (3 - (-4); (-1) - (-2); (-1) - 1) = (7; 1; -2)
Вектор KM:
KM = M - K = ((-4) - 2; (-2) - 1; 1 - (-3)) = (-6; -3; 4)
2)|MN| = sqrt((MNx)^2 + (MNy)^2 + (MNz)^2)
|MN| = sqrt((7)^2 + (1)^2 + (-2)^2) = sqrt(49 + 1 + 4) = sqrt(54) ≈ 7.35
3)PF = 3MN - 2KM
PFx = 3(MNx) - 2(KMx) = 3(7) - 2(-6) = 21 + 12 = 33
PFy = 3(MNy) - 2(KMy) = 3(1) - 2(-3) = 3 + 6 = 9
PFz = 3(MNz) - 2(KMz) = 3(-2) - 2(4) = -6 - 8 = -14
(33;9;-14)
3.
Mx = (Ax + Cx) / 2 = (3 + (-3)) / 2 = 0 / 2 = 0
My = (Ay + Cy) / 2 = (-5 + 9) / 2 = 4 / 2 = 2
Mz = (Az + Cz) / 2 = (0 + (-6)) / 2 = (-6) / 2 = -3
M дорівнюють (0, 2, -3).
Nx = (Bx + Cx) / 2 = (7 + (-3)) / 2 = 4 / 2 = 2
Ny = (By + Cy) / 2 = (1 + 9) / 2 = 10 / 2 = 5
Nz = (Bz + Cz) / 2 = (4 + (-6)) / 2 = -2 / 2 = -1
N дорівнюють (2, 5, -1).
d(MN) = sqrt((Nx - Mx)^2 + (Ny - My)^2 + (Nz - Mz)^2)
d(MN) = sqrt((2 - 0)^2 + (5 - 2)^2 + (-1 - (-3))^2)
= sqrt(2^2 + 3^2 + 2^2)
= sqrt(4 + 9 + 4)
= sqrt(17)
≈ 4.12
АВС приблизно дорівнює 4.12 одиницям.
4.
1)-2/1 = 8/-4 = -4/к
-2/1 = 8/-4
-2 * -4 = 1 * 8
8 = 8
2)a · b = (-2)(1) + (8)(-4) + (-4)(к) = -2 - 32 - 4к = 0
2 - 32 - 4к = 0
-34 - 4к = 0
-4к = 34
к = -34/4
к = -8.5
5.
Вектор (2а + Б)
(2а + Б) = (2 * 3 + (-1); 2 * (-1) + (-2); 2 * (-2) + 3) = (6 - 1; -2 - 2; -4 + 3) = (5; -4; -1)
Вектор (а - Б)
(а - Б) = (3 - (-1); -1 - (-2); -2 - 3) = (3 + 1; -1 + 2; -2 - 3) = (4; 1; -5)
--------------------------
(2а + Б) · (а - Б) = (5 * 4) + (-4 * 1) + (-1 * (-5)) = 20 - 4 + 5 = 21
добуток векторів (2а + Б) і (а - Б) дорівнює 21.
6.
Вектор AB: AB = B - A = (-1 - (-4); 5 - (-1); -1 - 4) = (3; 6; -5)
Вектор AC: AC = C - A = (3 - (-4); 3 - (-1); -1 - 4) = (7; 4; -5)
AB × AC = (ABy * ACz - ABz * ACy, ABz * ACx - ABx * ACz, ABx * ACy - ABy * ACx)
тобто
= (6 * (-5) - (-5) * 4, (-5) * 7 - 3 * (-5), 3 * 4 - 6 * 7)
= (-10, -20, -30)
|AB × AC| = sqrt((-10)^2 + (-20)^2 + (-30)^2) = sqrt(100 + 400 + 900) = sqrt(1400) ≈ 37.42
Площа трикутника АВС = |AB × AC| / 2 ≈ 37.42 / 2 ≈ 18.71
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Answers & Comments
1)Вектор MN:
MN = N - M = (3 - (-4); (-1) - (-2); (-1) - 1) = (7; 1; -2)
Вектор KM:
KM = M - K = ((-4) - 2; (-2) - 1; 1 - (-3)) = (-6; -3; 4)
2)|MN| = sqrt((MNx)^2 + (MNy)^2 + (MNz)^2)
|MN| = sqrt((7)^2 + (1)^2 + (-2)^2) = sqrt(49 + 1 + 4) = sqrt(54) ≈ 7.35
3)PF = 3MN - 2KM
PFx = 3(MNx) - 2(KMx) = 3(7) - 2(-6) = 21 + 12 = 33
PFy = 3(MNy) - 2(KMy) = 3(1) - 2(-3) = 3 + 6 = 9
PFz = 3(MNz) - 2(KMz) = 3(-2) - 2(4) = -6 - 8 = -14
(33;9;-14)
3.
Mx = (Ax + Cx) / 2 = (3 + (-3)) / 2 = 0 / 2 = 0
My = (Ay + Cy) / 2 = (-5 + 9) / 2 = 4 / 2 = 2
Mz = (Az + Cz) / 2 = (0 + (-6)) / 2 = (-6) / 2 = -3
M дорівнюють (0, 2, -3).
Nx = (Bx + Cx) / 2 = (7 + (-3)) / 2 = 4 / 2 = 2
Ny = (By + Cy) / 2 = (1 + 9) / 2 = 10 / 2 = 5
Nz = (Bz + Cz) / 2 = (4 + (-6)) / 2 = -2 / 2 = -1
N дорівнюють (2, 5, -1).
d(MN) = sqrt((Nx - Mx)^2 + (Ny - My)^2 + (Nz - Mz)^2)
d(MN) = sqrt((2 - 0)^2 + (5 - 2)^2 + (-1 - (-3))^2)
= sqrt(2^2 + 3^2 + 2^2)
= sqrt(4 + 9 + 4)
= sqrt(17)
≈ 4.12
АВС приблизно дорівнює 4.12 одиницям.
4.
1)-2/1 = 8/-4 = -4/к
-2/1 = 8/-4
-2 * -4 = 1 * 8
8 = 8
2)a · b = (-2)(1) + (8)(-4) + (-4)(к) = -2 - 32 - 4к = 0
2 - 32 - 4к = 0
-34 - 4к = 0
-4к = 34
к = -34/4
к = -8.5
5.
Вектор (2а + Б)
(2а + Б) = (2 * 3 + (-1); 2 * (-1) + (-2); 2 * (-2) + 3) = (6 - 1; -2 - 2; -4 + 3) = (5; -4; -1)
Вектор (а - Б)
(а - Б) = (3 - (-1); -1 - (-2); -2 - 3) = (3 + 1; -1 + 2; -2 - 3) = (4; 1; -5)
--------------------------
(2а + Б) · (а - Б) = (5 * 4) + (-4 * 1) + (-1 * (-5)) = 20 - 4 + 5 = 21
добуток векторів (2а + Б) і (а - Б) дорівнює 21.
6.
Вектор AB: AB = B - A = (-1 - (-4); 5 - (-1); -1 - 4) = (3; 6; -5)
Вектор AC: AC = C - A = (3 - (-4); 3 - (-1); -1 - 4) = (7; 4; -5)
AB × AC = (ABy * ACz - ABz * ACy, ABz * ACx - ABx * ACz, ABx * ACy - ABy * ACx)
тобто
= (6 * (-5) - (-5) * 4, (-5) * 7 - 3 * (-5), 3 * 4 - 6 * 7)
= (-10, -20, -30)
|AB × AC| = sqrt((-10)^2 + (-20)^2 + (-30)^2) = sqrt(100 + 400 + 900) = sqrt(1400) ≈ 37.42
Площа трикутника АВС = |AB × AC| / 2 ≈ 37.42 / 2 ≈ 18.71