what is teh angle between the vector i + 2j + 3k and the x axis

To find the angle between the vector \(\mathbf{v} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}\) and the \(x\)-axis, we use the dot product formula. 

Given:
- Vector \(\mathbf{v} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}\)
- Unit vector along the \(x\)-axis is \(\mathbf{i}\) (which can be written as \(\mathbf{i} + 0\mathbf{j} + 0\mathbf{k}\))

The dot product \(\mathbf{v} \cdot \mathbf{i}\) is given by:
\[
\mathbf{v} \cdot \mathbf{i} = (1)(1) + (2)(0) + (3)(0) = 1
\]

The magnitude of \(\mathbf{v}\) is:
\[
|\mathbf{v}| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14}
\]

The magnitude of \(\mathbf{i}\) is:
\[
|\mathbf{i}| = \sqrt{1^2 + 0^2 + 0^2} = 1
\]

Now, the angle \(\theta\) between \(\mathbf{v}\) and the \(x\)-axis is found using the dot product formula:
\[
\cos \theta = \frac{\mathbf{v} \cdot \mathbf{i}}{|\mathbf{v}| |\mathbf{i}|} = \frac{1}{\sqrt{14} \times 1} = \frac{1}{\sqrt{14}}
\]

So, the angle \(\theta\) is:
\[
\theta = \cos^{-1}\left(\frac{1}{\sqrt{14}}\right)
\]

This is the exact value of the angle. To express it as a decimal:
\[
\theta \approx \cos^{-1}\left(0.2673\right) \approx 74.5^\circ
\]

### Summary:
The angle between the vector \(\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}\) and the \(x\)-axis is approximately **74.5 degrees**.

Would you like more details or have any questions? Here are some related questions to explore:

1. What is the angle between the vector and the \(y\)-axis?
2. How would you find the angle between two arbitrary vectors?
3. Can the dot product be negative, and what does that signify?
4. What are some properties of the dot product of two vectors?
5. How would you calculate the projection of one vector onto another?

**Tip:** When calculating angles between vectors, always ensure both vectors are expressed in the same coordinate system!

What is the angle between the vector i + 2j + 3k and the x-axis?

Learn how to calculate the angle between the vector i + 2j + 3k and the x-axis using the dot product formula. This problem involves vector mathematics, including calculating magnitudes and using trigonometric identities. Suitable for students in Grades 10-12.

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