Compute the dot product and angle between the two vectors in degrees.

**u** = <3,5>, **v** = <2,4>

Round to three decimal places when necessary.

### Step 1: Calculate the Dot Product of the Vectors
The dot product of two vectors \(\mathbf{u} = \langle u_1, u_2 \rangle\) and \(\mathbf{v} = \langle v_1, v_2 \rangle\) is given by:

\[
\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2
\]

Substitute the values for \(\mathbf{u} = \langle 3, 5 \rangle\) and \(\mathbf{v} = \langle 2, 4 \rangle\):

\[
\mathbf{u} \cdot \mathbf{v} = (3)(2) + (5)(4) = 6 + 20 = 26
\]

### Step 2: Compute the Magnitudes of Each Vector
The magnitude of a vector \(\mathbf{u} = \langle u_1, u_2 \rangle\) is calculated by:

\[
\|\mathbf{u}\| = \sqrt{u_1^2 + u_2^2}
\]

For vector \(\mathbf{u} = \langle 3, 5 \rangle\):

\[
\|\mathbf{u}\| = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34} \approx 5.831
\]

For vector \(\mathbf{v} = \langle 2, 4 \rangle\):

\[
\|\mathbf{v}\| = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} \approx 4.472
\]

### Step 3: Calculate the Angle Between the Vectors
The angle \(\theta\) between two vectors is given by:

\[
\cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|}
\]

Substitute the values:

\[
\cos \theta = \frac{26}{5.831 \times 4.472} = \frac{26}{26.070} \approx 0.997
\]

To find the angle in degrees:

\[
\theta = \cos^{-1}(0.997) \approx 4.122^\circ
\]

### Final Answer:
- **Dot product**: 26
- **Angle**: \(4.122^\circ\)

Would you like any more details or have questions?

Here are 5 related questions:
1. How do you find the cross product of two 3D vectors?
2. What is the significance of the dot product in physics?
3. How do we compute angles between vectors in higher dimensions?
4. Can the dot product be negative? What does that signify?
5. How is the cosine of an angle used to determine vector similarity?

**Tip**: The dot product is often used in projections, allowing us to find how much one vector contributes to another in terms of direction and magnitude.

Compute the dot product and angle between the two vectors in degrees. **u** = <3,5>, **v** = <2,4>. Round to three decimal places when necessary.

Learn how to compute the dot product and the angle between the vectors u = <3,5> and v = <2,4>. The detailed solution covers vector algebra concepts, the dot product formula, vector magnitudes, and trigonometry to calculate angles. Suitable for students in Grades 10-12.

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