Rounding to three decimal places, the unit tangent vector is .
Steps
Step 1 :The level curve of a function is defined by the equation for some constant . The gradient of at a point is perpendicular to the level curve at that point. Therefore, the tangent to the level curve at is perpendicular to the gradient of at .
Step 2 :First, we need to find the gradient of . The gradient of is a vector whose components are the partial derivatives of with respect to and . So we need to compute the partial derivatives and .
Step 3 :The partial derivative of with respect to is . Evaluating this at the point gives .
Step 4 :The partial derivative of with respect to is . Evaluating this at the point gives .
Step 5 :Therefore, the gradient of at the point is .
Step 6 :The tangent vector to the level curve at is perpendicular to the gradient vector. Therefore, we can find the tangent vector by rotating the gradient vector by 90 degrees. This can be done by swapping the components of the gradient vector and changing the sign of one of them. We want the tangent vector to have a positive component, so we change the sign of the component. This gives the tangent vector .
Step 7 :Finally, we need to normalize the tangent vector to make it a unit vector. The length of the vector is . Therefore, the unit tangent vector is .
Step 8 :Rounding to three decimal places, the unit tangent vector is .