Answer to Question #241630 in Analytic Geometry for Bholu

Question #241630

(1) a) Give parametric equation (point-direction form) of the line which lies on both of the planes: x+y+z= 1 and -x+2y + 10z 2. What is the direction d of this line? b) Let ny and n₂ be the normal vectors to the two given planes. Without actual computation, describe the relationship between d and n₁ x n₂.


1
Expert's answer
2021-09-26T18:11:35-0400

a) n1=1,1,1,n2=1,2,10n_1=\langle 1, 1, 1\rangle, n_2=\langle -1, 2, 10\rangle


n1×n2=ijk1111210n_1\times n_2=\begin{vmatrix} i & j & k \\ 1 & 1 & 1 \\ -1 & 2 & 10 \\ \end{vmatrix}

=i11210j11110+k1112=i\begin{vmatrix} 1 & 1 \\ 2 & 10 \end{vmatrix}-j\begin{vmatrix} 1 & 1 \\ -1 & 10 \end{vmatrix}+k\begin{vmatrix} 1 & 1 \\ -1 & 2 \end{vmatrix}

=8i11j+3k=8i-11j+3k

d=8,11,3d=\langle 8, -11, 3\rangle

x+y+z=1x+y+z= 1x+2y+10z=2-x+2y + 10z =2

x+y+z=1x+y+z= 13y+11z=33y+11z= 3

x1=0,y1=1,z1=0x_1=0, y_1=1, z_1=0


x2=8,y2=10,z2=3x_2=8, y_2=-10, z_2=3

d=x2x1,y2y1,z2z1d=\langle x_2-x_1, y_2-y_1, z_2-z_1\rangle

d=80,101,30d=\langle 8-0, -10-1, 3-0\rangle

d=8,11,3d=\langle 8, -11, 3\rangle

b)


d=n1×n2d=n_1\times n_2

dn1,dn2d\perp n_1, d\perp n_2


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