probability density function of a vector

hello group,
bold are vectors, unbold scalars.
I know that is a and b are nonzero scalars and X is a random variable with pdf p(x), then the pdf of Y=aX+b
is p(y)=p((x-b)/a)/|a|

I would like to know the above theorem when a and b are constant vectors but X is a scalar random variable
Given Y=aX+b, find p(Y)?

I assuumed that this formula that holds for scalars will work for vectors too
p(Y)=Int[-inf,inf]d(y-(a*x-b))p(x)dx

where d is the dirac delta, then I dont know if this is the appropiate way to interpret the dirac-delta in this problem.
Let Y=(Y1,Y2,Y3), a=(a1,a2,a3), and b=(b1,b2,b3)
d(y-(a*x+b))=d(y1-(a1*x+b1))d(y2-(a2*x+b2))d(y3-(a3*x+b3))

then using the property of the dirac-delta:
d(f(x))=Sum{i}d(x-xi)/|f'(xi)|
where xi are the zeros of f(x), f(xi)=0
Let f(x)=y1-(a1*x+b1), f'(x)=a1
xi=(y1-b1)/a1, only one zero
I put it back to get this result:

p(y)=d(y2-(a2*xi+b2))d(y3-(a3*xi+b3))p(xi)/|a1|, is it correct?