What Special Relationship Is Seen Between the Radius of the Circle & the Line Tangent to the Circle?

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Special relationships that are seen between the circle and the line tangent to the circle will require you to consider the radius. Find out what special relationship is seen between the circle and the line tangent to the circle with help from an experienced mathematics professional in this free video clip.

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Hello, my name is Walter Unglaub. And this is what is the special relationship between the radius of the circle and the line tangent to the circle. If we consider a circle here with a radius R and the line that is tangent to the circle. The important or special relationship between these two lines here, is that they're always 90 degrees with respect to one another. This means that if I change the angle of the radius vector in this case, the way I drew it. That the line tangent to that point, the angle between those two lines is going to be equal to 90 degrees. So, that is the special relationship. And borrowing a concept from Vector Calculus. If two vectors are perpendicular, now we know that the inner product will be equal to zero. So, if the slope of my radius vector here for example, is M sub R equal to negative one. And the slope of the tangent line is M sub T is equal to positive one. Then, I can write the inner product as R dot T. And this is equal to the magnitude of the radius vector times the magnitude of the tangent vector times cosine theta. If I don't know the angle, but I know the respective components to each of the vectors. Then, I can also calculate the inner product by simply using the vector or arrays. So, let's say that the radius vector has one as the X component, where I(hat) is the unit vector in the X direction. But then it has minus one times j(hat), where j(hat) is the unit vector in the vertical direction. And let's say, for our perpendicular tangent line, we have a slope of one. So, we have plus one in the X direction and then, plus one in the vertical Y direction. To calculate the inner product, I simply take the vector for a radius, which is going to be one, negative one. And multiply it in matrix fashion to the vector for the tangent line. And this is simply going to be equal to one times minus one time one, because of this negative sign. And this is simply equal to zero. So, indeed these two vectors are perpendicular to each other. And that simply means that the angle between them is 90 degrees. Or in other words, in terms of radians, it's going to be equal to Pie over two radians or rads for short. my name is Walter Unglaub, and this is what is the special relationship between the radius of the circle and the line tangent to the circle.

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