To find the integral of a vector function r(t)=(r(t)1)i+(r(t)2)j+(r(t)3)k, we simply replace each coefficient with its integral. In other words, the integral of the vector function comes in the same form, just with each coefficient replaced by its own integral.
Read MoreTo find the unit tangent vector for a vector function, we use the formula T(t)=(r'(t))/(||r'(t)||), where r'(t) is the derivative of the vector function and t is given. We’ll start by finding the derivative of the vector function, and then we’ll find the magnitude of the derivative. Those two values will give us everything we need in order to build the expression for the unit tangent vector.
Read MoreTo find curvature at a particular point, we’ll 1) Find r'(t) and use it to 2) Find |r'(t)| and then use r'(t) and |r'(t)| to 3) Find T(t), and then use it to 4) Find T'(t), and then use it to 5) Find |T'(t)|, and then use |r'(t)| and |T'(t)| to 6) Find curvature at the point t that we’re interested in.
Read MoreA vector field F is called conservative if it’s the gradient of some scalar function. In this situation f is called a potential function for F. In this lesson we’ll look at how to find the potential function for a vector field.
Read MoreIndependence of path is a property of conservative vector fields. If a conservative vector field contains the entire curve C, then the line integral over the curve C will be independent of path, because every line integral in a conservative vector field is independent of path, since all conservative vector fields are path independent.
Read MoreGreen’s Theorem gives us a way to change a line integral into a double integral. If a line integral is particularly difficult to evaluate, then using Green’s Theorem to change it to a double integral might be a good way to approach the problem.
Read MoreIf we want to find the acute angle between two lines, we can convert the lines to standard vector form and then use the formula cos(theta)=(a•b)/(|a||b|), where a and b are the given vectors, a•b is the dot product of the vectors, |a| is the magnitude of the vector a (its length) and |b| is the magnitude of the vector b (its length).
Read MoreVector, parametric, and symmetric equations are different types of equations that can be used to represent the same line. We use different equations at different times to tell us information about the line, so we need to know how to find all three types of equations.
Read MoreIn this lesson we’ll look at the step-by-step process for finding the equations of the normal and osculating planes of a vector function. We’ll need to use the binormal vector, but we can only find the binormal vector by using the unit tangent vector and unit normal vector, so we’ll need to start by first finding those unit vectors.
Read MoreThe unit tangent vector T(t) of a vector function is the vector that’s 1 unit long and tangent to the vector function at the point t. Remember that |r'(t)| is the magnitude of the derivative of the vector function at time t. The unit normal vector N(t) of the same vector function is the vector that’s 1 unit long and perpendicular to the unit tangent vector at the same point t.
Read MoreSometimes the easiest way to sketch a three-dimensional curve is to sketch its projections on the xy-, xz-, and yz-coordinate planes. Think about the projections of a curve as the shadows they cast against the coordinate planes.
Read MoreTo find the scalar equation of a line, we’ll use the formulas x=x_0+at, y=y_0+bt, and z=z_0+ct, where P_0(x_0,y_0,z_0) is a given point and v=(a,b,c) is the given vector. The vector may also be in the format v=ai+bj+ck.
Read MoreWe say that two vectors a and b are orthogonal if they are perpendicular (their dot product is 0), parallel if they point in exactly the same or opposite directions, and never cross each other, otherwise, they are neither orthogonal or parallel. Since it’s easy to take a dot product, it’s a good idea to get in the habit of testing the vectors to see whether they’re orthogonal, and then if they’re not, testing to see whether they’re parallel.
Read MoreWhen we’re given two vectors with the same initial point, and they’re different lengths and pointing in different directions, we can think about each of them as a force. The longer the vector, the more force it pulls in its direction.
Oftentimes we want to be able to find the net force of the two vectors, which will be a third vector that counterbalances the force and direction of the other two. Think about the resultant vector as representing the amount of force and the direction in which you’d have to pull to cancel out the force from the other two vectors.
Read MoreAt any given point along a curve, we can find the acceleration vector ‘a’ that represents acceleration at that point. If we find the unit tangent vector T and the unit normal vector N at the same point, then the tangential component of acceleration a_T and the normal component of acceleration a_N are shown in the diagram below.
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