Question

If a line m passes through the point (−4,2,−3) and is parallel to line n given by the vector equation, then the vector equation of line m is:

a) r=⟨−4,2,−3⟩+t⟨a,b,c⟩
b) r=⟨−4,2,−3⟩+t⟨a,b,c⟩+s⟨p,q,r⟩
c) r=⟨a,b,c⟩+t⟨−4,2,−3⟩
d) r=⟨−4,2,−3⟩+t⟨−a,−b,−c⟩

Answer 1

The correct vector equation for line m, which is parallel to line n and passes through the point (-4,2,-3), is r = <-4, 2, -3> + t.

Step-by-step explanation:

If a line m passes through the point (-4,2,-3) and is parallel to a line n, then it must have the same direction vector as line n. The vector equation of a line in three-dimensional space can be written as r = ⟨x0, y0, z0⟩ + t⟨a, b, c⟩, where ⟨x0, y0, z0⟩ is any point on the line and ⟨a, b, c⟩ is a direction vector that indicates the direction of the line. Because line m is parallel to line n, their direction vectors must be the same (or proportional). So, if line n is given by the vector equation with direction vector ⟨a, b, c⟩, then the correct vector equation for line m, passing through (-4,2,-3), is r = ⟨-4, 2, -3⟩ + t⟨a, b, c⟩.