Would the vector equation of a line through the point (2,5) with direction vector v = (3, 2) be (2, 5) + t(3, 2)?
yes
Yes, that is correct. The vector equation of a line through a given point with a given direction vector is expressed as (x, y) = (a, b) + t(v₁, v₂), where (a, b) is the given point and (v₁, v₂) is the given direction vector. In this case, the point is (2, 5) and the direction vector is (3, 2), so the vector equation would indeed be (2, 5) + t(3, 2).
Yes, the vector equation of a line passing through a point, (2, 5), with a direction vector, v = (3, 2), can be written as (2, 5) + t(3, 2), where t is a scalar.
To understand why this is the case, let's break it down:
1. A point on the line: The given point (2, 5) is a point that lies on the line.
2. Direction vector: The direction vector, v = (3, 2), represents the direction of the line. For every value of t, if we multiply the direction vector by t and add it to the point (2, 5), we will reach a point on the line.
So, (2, 5) + t(3, 2) represents a point on the line that is obtained by starting from the point (2, 5) and moving in the direction of vector v = (3, 2) for a distance of t units.
In other words, as t varies, the equation (2, 5) + t(3, 2) generates all the points that lie on the line passing through (2, 5) and has a direction vector of (3, 2).