We are tasked with finding the tangential and normal components of the acceleration vector for the given position vector:

$r(t) = t \hat{i} + 2e^t \hat{j} + e^{2t} \hat{k}$

Step 1: Velocity Vector

The velocity vector $v(t)$ is the first derivative of the position vector $r(t)$:

$v(t) = \frac{d}{dt} r(t)$

For each component:

$v(t) = \frac{d}{dt} \left( t \hat{i} + 2e^t \hat{j} + e^{2t} \hat{k} \right)$ $v(t) = 1 \hat{i} + 2e^t \hat{j} + 2e^{2t} \hat{k}$

Step 2: Acceleration Vector

The acceleration vector $a(t)$ is the derivative of the velocity vector $v(t)$:

$a(t) = \frac{d}{dt} v(t)$

For each component:

$a(t) = \frac{d}{dt} \left( 1 \hat{i} + 2e^t \hat{j} + 2e^{2t} \hat{k} \right)$ $a(t) = 0 \hat{i} + 2e^t \hat{j} + 4e^{2t} \hat{k}$

Step 3: Magnitude of the Velocity

The magnitude of the velocity vector $v(t)$, which is necessary for calculating the tangential component, is:

$|v(t)| = \sqrt{(1)^2 + (2e^t)^2 + (2e^{2t})^2}$ $|v(t)| = \sqrt{1 + 4e^{2t} + 4e^{4t}}$

Step 4: Tangential and Normal Components of Acceleration

• Tangential component $a_T(t)$ is the component of acceleration in the direction of the velocity vector, calculated as:

$a_T(t) = \frac{d}{dt} |v(t)|$

• Normal component $a_N(t)$ is the component of acceleration perpendicular to the velocity vector, calculated as:

$a_N(t) = \sqrt{|a(t)|^2 - a_T(t)^2}$

Now, I'll proceed to calculate these components.The tangential and normal components of the acceleration vector are:

1. Tangential Component $a_T(t)$: $a_T(t) = \frac{8e^{4t} + 4e^{2t}}{\sqrt{4e^{4t} + 4e^{2t} + 1}}$

2. Normal Component $a_N(t)$: $a_N(t) = 2e^t$

Would you like further details on any step, or help with another question?

Here are five related questions for practice:

1. How do you compute the velocity vector for a different parameterized curve?
2. What is the significance of the tangential component in motion analysis?
3. How do normal and tangential components relate to circular motion?
4. How do you interpret the magnitude of acceleration geometrically?
5. What are the physical units of normal and tangential components?

Tip: When analyzing motion in 3D, always remember that the normal component of acceleration governs changes in the direction of velocity, while the tangential component handles changes in speed.