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Single Variable Calculus 

MultiVariable Calculus 

Acceleration Vector 
Arc Length (Vector Functions) 
Arc Length Function 
Arc Length Parameter 
Conservative Vector Fields 
Cross Product 
Curl 
Curvature 
Cylindrical Coordinates 
Lagrange Multipliers 
Line Integrals 
Partial Derivatives 
Partial Integrals 
Path Integrals 
Potential Functions 
Principal Unit Normal Vector 
Differential Equations 

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.
 
Help Keep 17Calculus Free 

In order to discuss curvature and a few other topics, we need to define a special vector called the unit tangent vector. As the name indicates, the unit tangent vector is a vector that is tangent to the curve and it's length is one.
What may not be obvious is that there is only one unit tangent vector and it points in the direction of motion. Given the vector function \(\vec{r}(t)\), the most basic equation we use to find the unit tangent vector is
\(\displaystyle{ \vec{T}(t) = \frac{\vec{r}'(t)}{ \ \vec{r}'(t) \ } }\)
The vector function \( \vec{r}(t) \) is often a position vector. As you know from basic calculus, the derivative of the position is velocity. So you will often see \(\vec{v}(t)=\vec{r}'(t)\) where \(\vec{v}(t)\) is referred to as the velocity vector. This allows us to write the unit tangent vector as \(\displaystyle{ \vec{T}(t) = \frac{\vec{v}(t)}{ \ \vec{v}(t) \ } }\).
This unit tangent vector is used a lot when calculating the principal unit normal vector, acceleration vector components and curvature. So take a few minutes to work some practice problems before going on to the next topic.
next: principal unit normal vector →

Conversion Between ABC Level (or 123) and New Numbered Practice Problems 

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. So, Practice A01 (1) is probably the first basic practice problem, A02 (2) is probably the second basic practice problem, etc. Practice B01 is probably the first intermediate practice problem and so on. 
Instructions   Unless otherwise instructed, calculate the unit tangent vector for the given vector function at the given point. If no point is given, find the general unit tangent vector \( \vec{T}(t) \).
\( \vec{r}(t) = t\vhat{i} + (1/t)\vhat{j} \), \(t=1\)
Problem Statement 

Calculate the unit tangent vector for the vector function \( \vec{r}(t) = t\vhat{i} + (1/t)\vhat{j} \) at the point \(t=1\).
Final Answer 

\(\displaystyle{ \vec{T}(1) = \frac{\vhat{i}  \vhat{j}}{\sqrt{2}} }\) 
Problem Statement 

Calculate the unit tangent vector for the vector function \( \vec{r}(t) = t\vhat{i} + (1/t)\vhat{j} \) at the point \(t=1\).
Solution 

\( \vec{r}'(t) = \vhat{i}  (1/t^2)\vhat{j} \)
\( \vec{r}'(1) = \vhat{i}  \vhat{j} \)
\(\displaystyle{ \vec{T}(1) = \frac{\vec{r}'(1)}{\ \vec{r}'(1) \} = \frac{\vhat{i}\vhat{j}}{\sqrt{2}} }\)
Here is a plot of the solution. The black line is the curve \( \vec{r}(t) \) with the black arrow indicating that it is being traced out left to right (or down the curve). (We have plotted only a section of the curve from x=1/4 to x=4.) The red vector is the unit tangent vector. Notice that it is pointing in the direction that the curve is being traced.
Final Answer 

\(\displaystyle{ \vec{T}(1) = \frac{\vhat{i}  \vhat{j}}{\sqrt{2}} }\) 
close solution 
\( \vec{r}(t) = \cos t \vhat{i} + 3t\vhat{j} + \) \( 2\sin 2t \vhat{k} \), \(t=0\)
Problem Statement 

Calculate the unit tangent vector for the vector function \( \vec{r}(t) = \cos t \vhat{i} + 3t\vhat{j} + \) \( 2\sin 2t \vhat{k} \) at the point \(t=0\).
Final Answer 

\( \vec{T}(0) = (3/5)\vhat{j} + (4/5)\vhat{k} \) 
Problem Statement 

Calculate the unit tangent vector for the vector function \( \vec{r}(t) = \cos t \vhat{i} + 3t\vhat{j} + \) \( 2\sin 2t \vhat{k} \) at the point \(t=0\).
Solution 

video by Krista King Math
Final Answer 

\( \vec{T}(0) = (3/5)\vhat{j} + (4/5)\vhat{k} \) 
close solution 
\( \vec{r}(t) = \langle 2\sin(t), 4\cos(t), 4\sin^2(t) \rangle, t=\pi/6\)
Problem Statement 

Calculate the unit tangent vector for the vector function \( \vec{r}(t) = \langle 2\sin(t), 4\cos(t), 4\sin^2(t) \rangle \) at the point \(t=\pi/6\).
Final Answer 

\(\displaystyle{ \frac{1}{\sqrt{19}} \langle \sqrt{3}, 2, 2\sqrt{3} \rangle }\) 
Problem Statement 

Calculate the unit tangent vector for the vector function \( \vec{r}(t) = \langle 2\sin(t), 4\cos(t), 4\sin^2(t) \rangle \) at the point \(t=\pi/6\).
Solution 

video by MIP4U
Final Answer 

\(\displaystyle{ \frac{1}{\sqrt{19}} \langle \sqrt{3}, 2, 2\sqrt{3} \rangle }\) 
close solution 
\( \vec{r}(t) = \langle t^3,2t^2 \rangle, t=1 \)
Problem Statement 

Calculate the unit tangent vector for the vector function \(\vec{r}(t) = \langle t^3,2t^2 \rangle \) at the point \(t=1\).
Final Answer 

\( \langle 3/5,4/5 \rangle \) 
Problem Statement 

Calculate the unit tangent vector for the vector function \(\vec{r}(t) = \langle t^3,2t^2 \rangle \) at the point \(t=1\).
Solution 

video by MIP4U
Final Answer 

\( \langle 3/5,4/5 \rangle \) 
close solution 
Find \(\vec{T}(t)\) and \(\vec{T}(0)\) for \(\vec{r}(t)=\langle 5t^2+1, e^{3t}, 2\sin(3t) \rangle \).
Problem Statement 

Find \(\vec{T}(t)\) and \(\vec{T}(0)\) for \(\vec{r}(t)=\langle 5t^2+1, e^{3t}, 2\sin(3t) \rangle \).
Final Answer 

\(\displaystyle{ \vec{T}(t) = \frac{\langle 10t, 3e^{3t}, 6\cos(3t) \rangle}{\sqrt{100t^2+9e^{6t}+36\cos^2(3t)}} }\) 
Problem Statement 

Find \(\vec{T}(t)\) and \(\vec{T}(0)\) for \(\vec{r}(t)=\langle 5t^2+1, e^{3t}, 2\sin(3t) \rangle \).
Solution 

video by MIP4U
Final Answer 

\(\displaystyle{ \vec{T}(t) = \frac{\langle 10t, 3e^{3t}, 6\cos(3t) \rangle}{\sqrt{100t^2+9e^{6t}+36\cos^2(3t)}} }\) 
close solution 
\( \vec{r}(t) = (t^3+t)\vhat{i} + \) \( (\ln(t^2))\vhat{j} + (\cos(\pi t))\vhat{k} \), \(t=1\)
Problem Statement 

Calculate the unit tangent vector for the vector function \( \vec{r}(t) = (t^3+t)\vhat{i} + \) \( (\ln(t^2))\vhat{j} + (\cos(\pi t))\vhat{k} \) at the point \( t=1 \).
Final Answer 

\( \langle \sqrt{2}/2, \sqrt{2}/2, 0 \rangle \) 
Problem Statement 

Calculate the unit tangent vector for the vector function \( \vec{r}(t) = (t^3+t)\vhat{i} + \) \( (\ln(t^2))\vhat{j} + (\cos(\pi t))\vhat{k} \) at the point \( t=1 \).
Solution 

video by PatrickJMT
Final Answer 

\( \langle \sqrt{2}/2, \sqrt{2}/2, 0 \rangle \) 
close solution 
\( \vec{r}(t) = \langle t\sqrt{2}, e^t, e^{t} \rangle \), \(t=0\)
Problem Statement 

Calculate the unit tangent vector for the vector function \( \vec{r}(t) = \langle t\sqrt{2}, e^t, e^{t} \rangle \) at the point \( t=0 \).
Final Answer 

\( \vec{T}(0) = \langle \sqrt{2}/2, 1/2, 1/2 \rangle \) 
Problem Statement 

Calculate the unit tangent vector for the vector function \( \vec{r}(t) = \langle t\sqrt{2}, e^t, e^{t} \rangle \) at the point \( t=0 \).
Solution 

video by David Lippman
Final Answer 

\( \vec{T}(0) = \langle \sqrt{2}/2, 1/2, 1/2 \rangle \) 
close solution 