Find the angle between the vectors V<4, -3, 1> and W<3, 2, -6>

To find the angle \(\theta\) between two vectors \(\mathbf{V}\) and \(\mathbf{W}\), we can use the dot product formula:

\[
\cos \theta = \frac{\mathbf{V} \cdot \mathbf{W}}{|\mathbf{V}| |\mathbf{W}|}
\]

1. **Calculate the dot product** \(\mathbf{V} \cdot \mathbf{W}\):

\[
\mathbf{V} \cdot \mathbf{W} = (4)(3) + (-3)(2) + (1)(-6) = 12 - 6 - 6 = 0
\]

2. **Calculate the magnitudes** of \(\mathbf{V}\) and \(\mathbf{W}\):

\[
|\mathbf{V}| = \sqrt{4^2 + (-3)^2 + 1^2} = \sqrt{16 + 9 + 1} = \sqrt{26}
\]

\[
|\mathbf{W}| = \sqrt{3^2 + 2^2 + (-6)^2} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7
\]

3. **Use the formula** to find \(\cos \theta\):

\[
\cos \theta = \frac{0}{\sqrt{26} \times 7} = 0
\]

4. **Determine the angle** \(\theta\):

\[
\theta = \cos^{-1}(0) = 90^\circ
\]

So, the angle between the vectors \(\mathbf{V}\) and \(\mathbf{W}\) is \(90^\circ\).

Would you like any more details or have any other questions?

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Here are 5 related questions for further exploration:

1. How do you calculate the dot product of two vectors?
2. What does it mean when the dot product of two vectors is zero?
3. How can you find the magnitude of a vector in three-dimensional space?
4. What is the geometric interpretation of the angle between two vectors?
5. How does the cross product differ from the dot product?

**Tip:** The dot product being zero indicates that the vectors are perpendicular to each other.

Learn how to find the angle between vectors V<4, -3, 1> and W<3, 2, -6> using the dot product formula. Follow a detailed step-by-step solution with calculations and explanations. Suitable for high school students learning vectors and trigonometry.

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