Give a vector parametric equation for the line through the point (−4,3)(−4,3) that is perpendicular to the line ⟨3+2𝑡,4+2𝑡

The line that is perpendicular to the line ⟨3+2t,4+2t⟩ has a direction vector that is the negative reciprocal of the direction vector of the given line. The direction vector of the given line is ⟨2,2⟩, so the direction vector of the line we are looking for is ⟨-1,1⟩ (since the negative reciprocal of 2/2 is -1/1).

The vector parametric equation of a line is given by:

r(t) = p + td

where p is a point on the line, d is the direction vector, and t is the parameter.

We are given that the line passes through the point (-4,3), so p = ⟨-4,3⟩.

Substituting p and d into the equation gives:

r(t) = ⟨-4,3⟩ + t⟨-1,1⟩ = ⟨-4-t, 3+t⟩

So, the vector parametric equation for the line through the point (-4,3) that is perpendicular to the line ⟨3+2t,4+2t⟩ is r(t) = ⟨-4-t, 3+t⟩.