. (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.
a)
If is orthogonal to then, dot product of and will be zero.
(P1) ⃗.
Here let be the vector along the line
let be the vector formed from the coefficient (a,b) of the line equation .
now
hence proved that and are orthogonal
b)
To prove formed from the coefficients of the equation is orthogonal to the plane,
is orthogonal to entry vector in the plane .
is a homogeneous equation.
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