Part 1: Finding the Magnitude of the Pulling Force

The pulling force is given as $\mathbf{F} = (94.0 \, \hat{i} + 133 \, \hat{j} + 34.0 \, \hat{k}) \, \text{N}$. To calculate the magnitude of the force, we use the formula for the magnitude of a vector:

$|\mathbf{F}| = \sqrt{F_x^2 + F_y^2 + F_z^2}$

where $F_x = 94.0$, $F_y = 133$, and $F_z = 34.0$. Let's calculate the magnitude:

$|\mathbf{F}| = \sqrt{(94.0)^2 + (133)^2 + (34.0)^2}$

$|\mathbf{F}| = \sqrt{8836 + 17689 + 1156}$

$|\mathbf{F}| = \sqrt{27681}$

$|\mathbf{F}| \approx 166.35 \, \text{N}$

So, the magnitude of the pulling force is approximately 166.35 N.

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Part 2: Finding the Angle with the Vertical

The vertical direction is along the $\hat{k}$ axis, so we want to find the angle $\theta$ that the leash makes with the vertical. The angle between two vectors is given by:

$\cos \theta = \frac{\mathbf{F} \cdot \mathbf{k}}{|\mathbf{F}| \cdot |\mathbf{k}|}$

Here, $\mathbf{k} = (0, 0, 1)$ and $\mathbf{F} = (94.0, 133, 34.0)$. The dot product $\mathbf{F} \cdot \mathbf{k}$ is simply the $z$-component of $\mathbf{F}$, which is 34.0.

Thus,

$\cos \theta = \frac{34.0}{|\mathbf{F}|}$

Substitute the magnitude of $|\mathbf{F}|$ from Part 1:

$\cos \theta = \frac{34.0}{166.35}$

$\cos \theta \approx 0.2044$

Now, take the inverse cosine to find the angle:

$\theta = \cos^{-1}(0.2044) \approx 78.2^\circ$

So, the angle that the leash makes with the vertical is approximately 78.2 degrees.

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Let me know if you'd like further details or have any questions. Here are 5 related questions to consider:

1. How would the magnitude of force change if each component doubled?
2. What if the leash were pulled horizontally, how would that affect the angle?
3. What happens to the pulling force's magnitude if the vertical component $k$ is zero?
4. How can we find the angle with the horizontal instead of the vertical?
5. What if frictional force acted against the pulling force?

Tip: When working with vectors, always remember that the dot product helps to find the projection of one vector in the direction of another.