唔識做M2數學,請幫幫忙!! (3)

[匿名]

2012-10-25 3:56 am

1. Given vector u = (-1, 1, 2) and vector v = (0, 1, 0).
a) u x v , and
b) Show that u x v is orthogonal to both u and v.

2. Given three points A=(2, -3, 4), B=(0, 1, 2), and C=(-1, 2, 0) in a 3-D coordinate systems. Find the area of the triangle with A, B, and C as vertices.

3. Solve x2 – 2x + 2 = 0. Express the roots in the form of a+bi where a and b are real numbers.

更新1:

上面有d字顯示唔到請轉睇呢張圖, thx!! http://i987.photobucket.com/albums/ae355/delaynopemore/3-10.jpg

回答 (1)

用戶10012

2012-10-25 3:48 pm

✔ 最佳答案

(1) u x v = | i j k |
| - 1 1 2 |
| 0 1 0 |
= - 2i - k
(b) ( u x v ) dot u = ( - 2, 0, -1) dot (-1, 1, 2) = (-2)(-1) + (0)(1) + (-1)(2) = 0 so
(u x v) is orthogonal to u.
(u x v) dot v = (-2, 0, -1) dot (0, 1, 0) = 0, so (u x v) is orthogonal to v.
(2)
AB = B - A = (0, 1, 2) - (2, -3, 4) = (-2, 4, - 2)
BC = B - C = (2, -3, 4) - (-1, 2, 0) = ( 3, - 5, 4)
Area of triangle ABC = (1/2) | AB| |BC| sin C = (1/2) | AB x BC |.
AB x BC = | i j k |
| - 2 4 - 2 |
| 3 - 5 4 |
= 6 i - 2 j - 2k
| AB x BC | = sqrt [6^2 + (-2)^2 + (-2)^2 ] = sqrt (36 + 4 + 4) = sqrt 44
= 2 sqrt 11
So area of triangle ABC = sqrt 11.
(3)
x = {2 +/- sqrt [4 - 4(1)(2)]}/2 = 1 +/- sqrt(- 4)/2 = 1 +/- sqrt (-1) = 1 +/- i
so a = b = 1.
Roots are 1 + i and 1 - i.

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