數學 向量問題 找體積和面積?

令 A = (1,1,1) , B = (2,1,1,) , C = (1,5,1) , T = (1,1,3) , 則:
TA = ( 0 , 0 , -2 )
TB = ( 1 , 0 , -2 )
TC = ( 0 , 4 , -2 )
AB = ( 1 , 0 , 0 )
AC = ( 0 , 4 , 0 )

TA．( TB × TC ) =
| 0 0 -2 |
| 1 0 -2 |
| 0 4 -2 |
= - 2*( 1*4 - 0*0 )
= - 8

利用三向量所張開形成的四面體公式:
V
= (1/6) * ∣ TA．( TB × TC ) ∣
= (1/6) * ∣ - 8 ∣
= 4/3

ΔABC
= (1/2) * ∣ ∣AB∣ * ∣AC∣ * sin∠BAC ∣
= (1/2) * ∣ AB × AC ∣
= (1/2) * ∣ ( 0 , 0 , 4 ) ∣
= (1/2) * 4
= 2

ΔTAB
= (1/2) * ∣ TA × TB ∣
= (1/2) * ∣ ( 0 , -2 , 0 ) ∣
= (1/2) * 2
= 1

ΔTBC
= (1/2) * ∣ TB × TC ∣
= (1/2) * ∣ ( 8 , 2 , 4 ) ∣
= (1/2) * √( 64 + 4 + 16 )
= (1/2) * 2√21
= √21

ΔTAC
= (1/2) * ∣ TA × TC ∣
= (1/2) * ∣ ( 8 , 0 , 0 ) ∣
= (1/2) * 8
= 4

Area
= ΔABC + ΔTAB + ΔTBC + ΔTAC
= 2 + 1 + √21 + 4
= 7 + √21

Ans: area = 7+√21 , volume = 4/3

註:
此題的體積因為形狀特殊,所以也可配合幾何方法求出 :
觀察這四點的座標, A, B, C 皆在 z = 1 的平面上; 而 T 在 z = 3 的平面上, 所以這四點形成一個"四角椎".
ΔABC = (1/2) * ∣ AB × AC ∣ = 2
高 = 3 - 1 = 2
四角椎體積 = (1/3) * 底面積 * 高 = (1/3)*2*2 = 4/3

Area: (1x2/2)+(1x4/2)+(2x4/2)+根號21=7+根號21
Volume: (1x2/2)x4/3=4/3