Determine parametric equations for the line through (-2, 3) and parallel to the line with vector equation Vector r = (-2, 1) + t(6, 4).
Please explain how to do this!
just change the starting point.
r = (-2,3) + t(6,4)
same direction numbers, right?
Well, first of all, let's start by finding the direction vector of the line we want to find parametric equations for. The line we want is parallel to the line with vector equation Vector r = (-2, 1) + t(6, 4).
The direction vector of the line is the coefficient of t, which is (6, 4).
Now, let's use the given point (-2, 3) and the direction vector (6, 4) to find the parametric equations.
To do this, we can assign initial values for x, y, and z based on the given point (-2, 3). Let's assign x = -2 and y = 3.
Now, we can write the parametric equations as:
x = -2 + 6t
y = 3 + 4t
And that's it! The parametric equations for the line through (-2, 3) and parallel to the line with vector equation Vector r = (-2, 1) + t(6, 4) are x = -2 + 6t and y = 3 + 4t.
Hope that helps! If you need any more assistance, feel free to ask!
To determine parametric equations for a line parallel to another line, we need to find a direction vector for the line.
Given a line with the vector equation r = (-2, 1) + t(6, 4), the direction vector of this line is (6, 4).
Since we want to find a line parallel to this line, the direction vector for our new line will also be (6, 4).
Now, we need to find the parametric equations for the line passing through the point (-2, 3) with the direction vector (6, 4).
Let the parametric equations for the line be:
x = -2 + 6t
y = 3 + 4t
In these equations, x and y represent the coordinates of any point on the line, and t is the parameter that determines the position of the point on the line.
These equations represent the line passing through (-2, 3) and parallel to the line with the vector equation r = (-2, 1) + t(6, 4).
To determine the parametric equations for the line through point (-2, 3) and parallel to the line with vector equation r = (-2, 1) + t(6, 4), we need to find a vector that is parallel to the given line.
The given line has a direction vector of (6, 4). Any multiple of this vector will also be parallel to the line. Let's call the multiple "k".
So, a vector parallel to the given line is k(6, 4).
To find the specific value of "k", we can use the fact that the line we want to find passes through the point (-2, 3). We can set up an equation by substituting the coordinates of (-2, 3) into the vector equation:
(-2, 3) + t(k(6, 4))
Expanding this equation, we get:
(-2 + 6kt, 3 + 4kt)
We want the line to pass through (-2, 3) for any value of t. Therefore, the coefficients of t must cancel each other out. This means:
-2 + 6kt = -2
3 + 4kt = 3
Solving these equations, we get k = 0.
Therefore, the line through (-2, 3) and parallel to the line with vector equation r = (-2, 1) + t(6, 4) has the parametric equations:
x = -2
y = 3
Alternatively, you could also say:
x = -2 + 0t
y = 3 + 0t
Note that the parameter "t" is not present in the equations because the line is parallel to the given line, and hence it does not change as "t" varies.