Vectors and Parametric Equations
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
We can use the vector equation of a line to describe the motion along a line. We can also rewrite these equations to represent the motion along a line as a parametric equation. In order to create a vector equation or a parametric equation to describe the motion along a line we need to determine a starting position for time zero and a velocity.
I want to talk about motion on a straight line. Imagine an object that moves with constant velocity v at t=0 it's located at point 5 negative 4 and at t=4 it's at point negative 7, 12. It's very important that you know that if an object moves with constant velocity it will move in a straight line and that means that it's movement can be modeled by the vector equation for a line r=r0+tv now in this case r0 is the initial point when t=0 we get 5 negative 4. So r0 is the position vector for that point 5 negative 4 now we also know that when t=4 the position is negative 7 12 that means we get negative 7 12 equals r0, 5 negative 4 plus when t=4, 4 times the velocity whatever that might be.
And we know it's constant but we caould use this equation to find out exactly what it is, so let me subtract this vector from both sides and remember when you subtract vectors you subtract component wise so negative 7 minus 5 is negative 12, 12 minus negative 4 is 12+4 16=4v so I just divide both sides by 4 and I get that v is negative 3 4 that's the velocity and that means that the vector equation for the object's position is r=r0 remember r0 was 5 negative 4 plus t times the velocity vector which was negative 3 4.
Now it's more usual to see this written xy=5 negative 4+t times negative 3 4, this is the vector equation for the objects position but we can express it another way. You can take the right hand side and distribute the t over this vector and you get xy=5 negative 4 plus negative 3t 4t. And then you can add these 2 vectors and you get adding components wise 5-3t and negative 4 plus 4t but remember that 2 vectors are only equal if their components are equal and that means we get 2 equations x=5-3t and y equals negative 4+4t. These 2 equations also describe the position of the object and they're called parametric equations. We'll be studying parametric equations in this lesson.