find angle between vector a = 1i+3j and vector b = 2i+4j

To find the angle \(\theta\) between two vectors \(\mathbf{a} = \mathbf{i} + 3\mathbf{j}\) and \(\mathbf{b} = 2\mathbf{i} + 4\mathbf{j}\), you can use the dot product formula:

\[
\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}
\]

### Step 1: Calculate the dot product \(\mathbf{a} \cdot \mathbf{b}\)

\[
\mathbf{a} \cdot \mathbf{b} = (1)(2) + (3)(4) = 2 + 12 = 14
\]

### Step 2: Calculate the magnitudes \(|\mathbf{a}|\) and \(|\mathbf{b}|\)

\[
|\mathbf{a}| = \sqrt{(1)^2 + (3)^2} = \sqrt{1 + 9} = \sqrt{10}
\]
\[
|\mathbf{b}| = \sqrt{(2)^2 + (4)^2} = \sqrt{4 + 16} = \sqrt{20}
\]

### Step 3: Plug these values into the cosine formula

\[
\cos \theta = \frac{14}{\sqrt{10} \times \sqrt{20}} = \frac{14}{\sqrt{200}} = \frac{14}{10\sqrt{2}} = \frac{7}{5\sqrt{2}} = \frac{7\sqrt{2}}{10}
\]

### Step 4: Find the angle \(\theta\) using the inverse cosine

\[
\theta = \cos^{-1}\left(\frac{7\sqrt{2}}{10}\right)
\]

This is the exact value of the angle between the vectors. You can compute it numerically if needed:

\[
\theta \approx \cos^{-1}(0.9899) \approx 8.13^\circ
\]

Would you like more details or have any questions?

Here are five related questions:
1. How do you find the angle between vectors in 3D space?
2. What is the geometric interpretation of the dot product?
3. How do you determine if two vectors are orthogonal?
4. Can the dot product be used to find the projection of one vector onto another?
5. What is the difference between the dot product and the cross product?

**Tip:** The dot product of two vectors is zero if they are perpendicular to each other.

Learn how to find the angle between two vectors using the dot product and cosine formula. This problem involves vectors a = 1i + 3j and b = 2i + 4j, and uses fundamental trigonometric functions. Suitable for high school students.

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