To find the slope of a vector equation, you first need to express the vector in terms of its components. Typically, a vector equation can be represented in the form r(t) = x(t)i + y(t)j, where r(t) is the position vector, and x(t) and y(t) are functions of a parameter t.
The slope of the line represented by the vector equation can be determined by taking the derivative of the functions with respect to the parameter t. Specifically, you will calculate the derivatives x'(t) and y'(t).
The slope m can be found using the formula:
m = rac{y'(t)}{x'(t)}
This gives you the rate of change of y with respect to x, which represents the slope of the vector at a particular point. Make sure that x'(t) to avoid division by zero, as this indicates a vertical slope.
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For example, if you have a vector equation such as r(t) = (2t)i + (3t^2)j, you would first compute the derivatives: x'(t) = 2 and y'(t) = 6t. Then, you can substitute these into the slope formula:
m = rac{6t}{2} = 3t
This means that the slope of the vector equation varies depending on the value of t.