Determine vector and parametric equations for the line through the point A(2, 5) with direction vector Vector m = (1, -3).
r = A + mt = (2 + t)i + (5 - 3t)j
To determine the vector equation for the line passing through point A(2, 5) with direction vector m = (1, -3), we can use the formula:
r = r0 + t * m
where r represents the position vector on the line, r0 represents the position vector of point A, t represents a real number parameter, and m represents the direction vector.
Given that point A(2, 5) has a position vector r0 = (2, 5), and the direction vector m = (1, -3), we can substitute these values into the formula:
r = (2, 5) + t * (1, -3)
This is the vector equation of the line.
To find the parametric equations of the line, we can separate the components:
x = 2 + t
y = 5 - 3t
These are the parametric equations of the line passing through point A(2, 5) with direction vector m = (1, -3).
To determine the vector equation and parametric equations for a line, we need the point on the line and the direction vector.
Given:
Point A(2, 5)
Direction vector m = (1, -3)
1. Vector Equation:
The vector equation of a line passing through point A with direction vector m is given by:
r = a + tb
where r is the position vector of any point on the line, a is the position vector of the given point A, t is a scalar parameter, and b is the direction vector.
Plugging in the values, we get:
r = <2, 5> + t<1, -3>
So, the vector equation for the line is:
r = <2 + t, 5 - 3t>
2. Parametric Equations:
The parametric equations for x, y, and z (if applicable) can be obtained by separating the components of the vector equation:
x = 2 + t
y = 5 - 3t
Therefore, the parametric equations for the line are:
x = 2 + t
y = 5 - 3t
Note: Since only a direction vector m is given, the line is assumed to be in the xy-plane. However, if a direction vector m with three components is given, the equation can be extended to include the z component by adding z = c + 0t, where c is the z-coordinate of the given point A.