Let's solve this problem step-by-step using the provided information.

We are given two vectors: $\vec{a} = 8.0\hat{i} + 8.0\hat{j} + 8.0\hat{k}$ $\vec{b} = 7.0\hat{i} + 8.0\hat{j} + 7.0\hat{k}$

We are asked to calculate the angle $\theta$ between these vectors using the scalar (dot) product definition.

Step 1: Definition of the Scalar Product

The dot product of two vectors $\vec{a} \cdot \vec{b}$ is given by: $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta$ It can also be written in component form: $\vec{a} \cdot \vec{b} = a_x b_x + a_y b_y + a_z b_z$

Step 2: Component Form Calculation

We know that:

• $\vec{a} = (8.0, 8.0, 8.0)$
• $\vec{b} = (7.0, 8.0, 7.0)$

The dot product is: $\vec{a} \cdot \vec{b} = (8.0)(7.0) + (8.0)(8.0) + (8.0)(7.0)$ $\vec{a} \cdot \vec{b} = 56 + 64 + 56 = 176$

Step 3: Magnitude of Vectors

Next, we find the magnitudes of $\vec{a}$ and $\vec{b}$.

For $\vec{a}$: $|\vec{a}| = \sqrt{(8.0)^2 + (8.0)^2 + (8.0)^2} = \sqrt{64 + 64 + 64} = \sqrt{192} \approx 13.86$

For $\vec{b}$: $|\vec{b}| = \sqrt{(7.0)^2 + (8.0)^2 + (7.0)^2} = \sqrt{49 + 64 + 49} = \sqrt{162} \approx 12.73$

Step 4: Calculate the Angle

Using the dot product formula: $\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}$ Substitute the values: $\cos \theta = \frac{176}{(13.86)(12.73)} \approx \frac{176}{176.42} \approx 0.9976$

Now, to find $\theta$: $\theta = \cos^{-1}(0.9976) \approx 4.82^\circ$

Thus, the angle between the two vectors is approximately $4.82^\circ$.

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. How would the angle change if one of the vectors had a zero component?
2. What happens if the dot product of two vectors is zero?
3. How does the cross product differ from the dot product?
4. Can the angle between two vectors exceed 90 degrees, and what does that imply?
5. How can this method be extended to 4-dimensional vectors?

Tip: Always check the magnitudes of vectors carefully when calculating angles to avoid small rounding errors.