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Comment: Your vector equation isn't an equation. An equation always has '=' in it.ttpp1124 said:
Mark44 said:
Yes,, I understand how you got the equation, but I was responding to your request for someone to verify that your work was correct. Here's what I said before:ttpp1124 said:
One of the things I've always liked about mathematics, especially at somewhat higher levels, is that it's not difficult to verify your work for yourself. What I said above was how you can do this.Mark44 said:
A vector is a mathematical object that represents a magnitude and direction, while a parametric equation is a set of equations that describe the relationship between two or more variables. In the context of calculus and vectors, a vector equation is often used to represent a line or curve in space, while a parametric equation is used to describe the coordinates of points along that line or curve.
To determine the vector equation of a line in three-dimensional space, you need to know a point on the line and the direction vector of the line. The vector equation can then be written as r = a + tb, where r is the position vector of any point on the line, a is the known point, t is a scalar parameter, and b is the direction vector.
To find the parametric equations of a curve, you first need to express the curve in vector form. This can be done by writing the curve as the sum of two or more vectors. Then, each component of the vectors can be expressed as a function of a parameter, typically denoted by t. These functions will form the parametric equations of the curve.
Yes, calculus can be used to find the vector and parametric equations of a curve. Calculus concepts such as derivatives and integrals can be applied to vector equations to find the slope, curvature, and other properties of the curve. These properties can then be used to determine the parametric equations of the curve.
To convert between vector and parametric equations, you can use the components of the vector equation to form the parametric equations. The coordinates of the points on the curve can then be expressed as functions of a parameter, which will form the parametric equations. Conversely, the parametric equations can be combined to form the vector equation by expressing each coordinate as a function of the parameter and then combining them into a single vector equation.
