PROFESSOR: Find the unit tangent vector for the vector valued function r of t equals parentheses t squared minus 3 close parentheses i plus open parentheses 2t plus 1 close parentheses j plus open parentheses t minus 2 close parentheses k. The equation for our unit tangent is capital T of t equals r prime of t divided by the magnitude of r prime of t. So for this given r of t, we need first find its antiderivative, and then find its magnitude. And then divide those two values.

So given this function, r prime of t equals 2ti plus 2j plus k. And the magnitude of r prime of t is equal to the square root of 2t as a quantity squared. So that would be 4t squared plus 2 squared or 4 plus 1 squared, which is 1.

And again, we're taking the i, j, and k components and squaring those and then summing them. Take the square root to find the magnitude. So that means that capital T of t is equal to 2 ti-- 1 ti plus 2j plus k divided by the square root of 4t squared plus 5. We can split this into three terms of 2t over the square root t squared plus 5 i plus 2 over that quantity square root of 4t squared plus 5 j plus 1 over the square root of 4t squared plus 5 k.