Given each scalar equation, write a vector equation and the parametric equations.
8x + 9y = -45
To convert the given scalar equation into a vector equation and parametric equations, you can follow these steps:
Step 1: Vector Equation
In the vector equation, we need to represent the variables x and y as components of a vector. Let's define vector variable r = [x, y].
The given scalar equation is 8x + 9y = -45. To convert it into a vector equation, we can write:
r = [x, y] = [t, s] (where t and s represent scalar parameters)
Now, we need to find the components of the vector equation using the given equation:
8x + 9y = -45
By rearranging terms, we can express x and y in terms of t and s:
8t + 9s = -45
Therefore, the vector equation is:
r = [t, s] = [t, (-45 - 8t) / 9]
Step 2: Parametric Equations
To obtain the parametric equations, we separate the vector equation into two individual equations, one for each component, x and y:
For x:
x = t
For y:
y = (-45 - 8t) / 9
These are the parametric equations associated with the given scalar equation.
To summarize:
Vector equation:
r = [x, y] = [t, (-45 - 8t) / 9]
Parametric equations:
x = t
y = (-45 - 8t) / 9