Find the angle between the force vector D = (2.00
i − 4.00
j + 1.00
k) N and the force vector G = (3.00
i + 4.00
j + 10.0
k) N.

To find the angle \( \theta \) between two vectors \( \mathbf{D} \) and \( \mathbf{G} \), we can use the following formula:

\[
\cos \theta = \frac{\mathbf{D} \cdot \mathbf{G}}{|\mathbf{D}| |\mathbf{G}|}
\]

Where:
- \( \mathbf{D} \cdot \mathbf{G} \) is the dot product of the vectors \( \mathbf{D} \) and \( \mathbf{G} \),
- \( |\mathbf{D}| \) is the magnitude of vector \( \mathbf{D} \),
- \( |\mathbf{G}| \) is the magnitude of vector \( \mathbf{G} \).

### Step 1: Find the dot product \( \mathbf{D} \cdot \mathbf{G} \)
The vectors are:
\[
\mathbf{D} = (2.00 \, \hat{i} - 4.00 \, \hat{j} + 1.00 \, \hat{k}) \, \text{N}
\]
\[
\mathbf{G} = (3.00 \, \hat{i} + 4.00 \, \hat{j} + 10.0 \, \hat{k}) \, \text{N}
\]

The dot product is calculated as:
\[
\mathbf{D} \cdot \mathbf{G} = (2.00)(3.00) + (-4.00)(4.00) + (1.00)(10.0)
\]

### Step 2: Find the magnitudes of \( \mathbf{D} \) and \( \mathbf{G} \)
The magnitudes are given by:
\[
|\mathbf{D}| = \sqrt{(2.00)^2 + (-4.00)^2 + (1.00)^2}
\]
\[
|\mathbf{G}| = \sqrt{(3.00)^2 + (4.00)^2 + (10.0)^2}
\]

### Step 3: Use the dot product and magnitudes to find \( \cos \theta \), then solve for \( \theta \).

I'll calculate this now.The angle \( \theta \) between the force vectors \( \mathbf{D} \) and \( \mathbf{G} \) is \( 90^\circ \). This result means that the two vectors are perpendicular to each other, as the dot product between them is \( 0 \), confirming that the vectors are orthogonal.

Do you want further details or have any questions?

Here are 5 related questions for further exploration:
1. What happens if the dot product is negative?
2. How can we use the cross product to find the direction of torque given a force and position vector?
3. What would the angle between two parallel vectors be?
4. How does the magnitude of a vector relate to its components?
5. How do you interpret a force vector in physics in terms of its components?

**Tip:** The dot product of two vectors can give insights into their relative orientation—whether they are aligned, opposite, or perpendicular.

Find the angle between the force vector D = (2.00 i − 4.00 j + 1.00 k) N and the force vector G = (3.00 i + 4.00 j + 10.0 k) N.

Discover how to find the angle between two force vectors using the dot product method. The problem involves calculating the dot product and magnitudes of vectors D = (2.00 i − 4.00 j + 1.00 k) N and G = (3.00 i + 4.00 j + 10.0 k) N to determine the angle between them. Suitable for advanced high school and college students.

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