Question #185570

. (a) The equation ax + by = 0 represents a line through the origin in R2. Show that the vector n1 = (a, b) formed from the coefficients of the equation is orthogonal to the line, that is, orthogonal to every vector along the line. (b) The equation ax + by + cz = 0 represents a plane through the origin in R3. Show that the vector n2 = (a, b, c) formed from the coefficients of the equation is orthogonal to the plane, that is, orthogonal to every vector that lies in the plane.


1
Expert's answer
2021-04-27T12:29:14-0400

a)

If (P1)(P1) ⃗ is orthogonal to (n1)(n1) ⃗ then, dot product of (P1)(P1) ⃗ and (n1)(n1) ⃗ will be zero.


(P1) ⃗.(n1)=0(n1) ⃗=0


Here let (P1)(P1) ⃗ be the vector along the line ax+by=0ax +by=0


ax=byax=-by


xy=ba\frac{x}{y}=\frac{-b}{a}


p=bi+ajp=-bi+aj


let (n1)(n1) ⃗ be the vector formed from the coefficient (a,b) of the line equation .


(n1)=ai+bj(n1) ⃗=ai+bj


now (P1).(n1)=(ba+aj).(ai+bj)(P1) ⃗.(n1) ⃗=(-ba+aj).(ai+bj)

=ba+ab=-ba+ab

=0=0

hence proved that (P1)(P1) ⃗ and (n1)(n1) ⃗ are orthogonal


b)

To prove (n2)=(a,b,c)(n2) ⃗=(a,b,c) formed from the coefficients of the equation is orthogonal to the plane,

(a,b,c).(x,y,z)=0(a,b,c).(x,y,z)=0


(n2).(x,y,z)=0(n2) ⃗.(x,y,z)=0


(n2)(n2) ⃗ is orthogonal to entry vector (x,y,z)(x,y,z) in the plane .


ax+by+cz=0ax+by+cz=0 is a homogeneous equation.




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