To find the tangential and normal components of acceleration for the space curve $\mathbf{r}(t)$, we need to follow these steps:

1. Tangential Component of Acceleration ( $a_T$ ):

The tangential component is the projection of the acceleration vector $\mathbf{a}(t)$ along the direction of the velocity vector $\mathbf{v}(t)$. It is given by the formula:

$a_T = \frac{\mathbf{v}(t) \cdot \mathbf{a}(t)}{|\mathbf{v}(t)|}$

2. Normal Component of Acceleration ( $a_N$ ):

The normal component is the part of the acceleration vector that is perpendicular to the velocity vector. It is calculated by:

$a_N = \frac{|\mathbf{v}(t) \times \mathbf{a}(t)|}{|\mathbf{v}(t)|}$

Given:

The position vector is $\mathbf{r}(t) = t\mathbf{i} + 3t\mathbf{j} - 6t\mathbf{k}$.

Let’s compute the tangential and normal components step by step.

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Step 1: Velocity Vector $\mathbf{v}(t)$

$\mathbf{v}(t) = \frac{d}{dt} \mathbf{r}(t) = \frac{d}{dt} (t\mathbf{i} + 3t\mathbf{j} - 6t\mathbf{k})$ $\mathbf{v}(t) = \mathbf{i} + 3\mathbf{j} - 6\mathbf{k}$

At $t = 3$, the velocity vector is: $\mathbf{v}(3) = \mathbf{i} + 3\mathbf{j} - 6\mathbf{k}$

Step 2: Acceleration Vector $\mathbf{a}(t)$

$\mathbf{a}(t) = \frac{d}{dt} \mathbf{v}(t)$ Since $\mathbf{v}(t)$ is constant, $\mathbf{a}(t) = 0$.

At $t = 3$, the acceleration vector is: $\mathbf{a}(3) = 0$

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Step 3: Tangential Component of Acceleration $a_T$

$a_T = \frac{\mathbf{v}(3) \cdot \mathbf{a}(3)}{|\mathbf{v}(3)|}$ Since $\mathbf{a}(3) = 0$, we have: $a_T = 0$

Step 4: Normal Component of Acceleration $a_N$

$a_N = \frac{|\mathbf{v}(3) \times \mathbf{a}(3)|}{|\mathbf{v}(3)|}$ Again, since $\mathbf{a}(3) = 0$, the cross product $\mathbf{v}(3) \times \mathbf{a}(3) = 0$. Therefore: $a_N = 0$

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Final Answer:

• Tangential Component $a_T$ = 0
• Normal Component $a_N$ = 0

Let me know if you need more details or have any questions!

Here are 5 related questions:

1. What is the magnitude of the velocity vector at $t = 3$?
2. How do we compute curvature for a space curve like $\mathbf{r}(t)$?
3. What is the direction of the velocity vector at $t = 3$?
4. How would acceleration behave for a non-constant velocity function?
5. What does it mean when the normal component of acceleration is zero?

Tip: The tangential acceleration reflects the change in speed, while the normal acceleration reflects the change in direction!