Yahoo奇摩知識+ 將於 2021 年 5 月 4 日 (美國東部時間) 終止服務。自 2021 年 4 月 20 日 (美國東部時間) 起,Yahoo奇摩知識+ 網站將會轉為唯讀模式。其他 Yahoo奇摩產品與服務或您的 Yahoo奇摩帳號都不會受影響。如需關於 Yahoo奇摩知識+ 停止服務以及下載您個人資料的資訊,請參閱說明網頁。

Loking 發問時間: 科學數學 · 1 0 年前

lines and planes in 3d

consider two planes

P1 : 3x-2y+4z=1

p2 2x-2y+z=3

and a line

L x=3+2t, y=1-5t, z=-2+6t

show that for any real number k, the plane

(2x-2y+4z-1)+k(2x-2y+z-3)=0 contains the line of intersection of P1 and P2

find an eqt of the plane containing the line of intersection of P1 and P2, and // L

find the shortest distance b/w L and the line of intersection of P1 and P2

已更新項目:

P1 : 3x-2y+4z=1 no typo

2 個已更新項目:

may u explain a bit more on how do you find the distance between a line and a plane which is parallel to it?

3 個已更新項目:

what is the method in general?

2 個解答

評分
  • 1 0 年前
    最佳解答

    1. If A(x,y,z) belongs to P1 and P2, then

    3x-2y+4z-1=0 and 2x-2y+z-3=0,

    so, (3x-2y+4z-1)+k(2x-2y+z-3)=0+k*0=0

    ie. A(x,y,z) satisfies (3x-2y+4z-1)+k(2x-2y+z-3)=0 for any k.

    2. Let plane E: (3x-2y+4z-1)+k(2x-2y+z-3)=0 parallel L, then

    the normal vector of E (3+2k, -2-2k, 4+k) is perpendicular to the

    direction vector (2, -5, 6) of L, thus, 2(3+2k)-5(-2-2k)+6(4+k)=0, k=-2.

    Hence, E : -x+2y+2z+5=0

    3. The sortest distance = d(E, L) = | -3+2-4+5|/3=0.

    2011-05-16 01:43:09 補充:

    3. The shortest distance = d(E, L)= d(E, A(3, 1, -2))= | -3 + 2 -4 + 5 |/3=0.

    題目: P1是 3x-2y+4z=1 or 2x-2y+4z=1?

    2011-05-16 02:17:37 補充:

    (2x-2y+4z-1)+k(2x-2y+z-3)=0 ????

    2011-05-16 02:20:12 補充:

    (2x-2y+4z-1)+k(2x-2y+z-3)=0 ????

    2011-05-16 18:17:23 補充:

    Because L//E, so the distance b/w E and L equals the distance b/w any point A of L and the plan E.

    Taking A(3,1,-2), thus the distance d(L, E)= d(A, E)= | -3+2 -4 +5| /√(1+4+4)= 0.

    In general, if E contains L2 and E parrelle to L1, then d(L1, L2)=d(E, L1)=d(E, A),

    where A is any point of L1.

  • Sam
    Lv 6
    1 0 年前

    To : 煩惱即是菩提 ( 知識長)

    第三題你是不是看錯題目?

還有問題?馬上發問,尋求解答。