9,0,-9     9,9,0  find angle between in terms of pi

To find the angle between the two vectors \( \mathbf{a} = (9, 0, -9) \) and \( \mathbf{b} = (9, 9, 0) \), we can use the formula for the angle between two vectors, which is given by:

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

### Step 1: Find the dot product \( \mathbf{a} \cdot \mathbf{b} \)
The dot product of two vectors \( \mathbf{a} = (a_1, a_2, a_3) \) and \( \mathbf{b} = (b_1, b_2, b_3) \) is:

\[
\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3
\]

Substituting the values from the given vectors:

\[
\mathbf{a} \cdot \mathbf{b} = (9)(9) + (0)(9) + (-9)(0) = 81
\]

### Step 2: Find the magnitudes of \( \mathbf{a} \) and \( \mathbf{b} \)

The magnitude of a vector \( \mathbf{a} = (a_1, a_2, a_3) \) is given by:

\[
|\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}
\]

For \( \mathbf{a} = (9, 0, -9) \):

\[
|\mathbf{a}| = \sqrt{9^2 + 0^2 + (-9)^2} = \sqrt{81 + 81} = \sqrt{162} = 9\sqrt{2}
\]

For \( \mathbf{b} = (9, 9, 0) \):

\[
|\mathbf{b}| = \sqrt{9^2 + 9^2 + 0^2} = \sqrt{81 + 81} = \sqrt{162} = 9\sqrt{2}
\]

### Step 3: Calculate \( \cos(\theta) \)

Now, we substitute the dot product and magnitudes into the formula for \( \cos(\theta) \):

\[
\cos(\theta) = \frac{81}{(9\sqrt{2})(9\sqrt{2})} = \frac{81}{162} = \frac{1}{2}
\]

### Step 4: Find \( \theta \)

We know that \( \cos(\theta) = \frac{1}{2} \), and the corresponding angle is:

\[
\theta = \frac{\pi}{3}
\]

So, the angle between the vectors \( \mathbf{a} = (9, 0, -9) \) and \( \mathbf{b} = (9, 9, 0) \) is \( \frac{\pi}{3} \) radians.

Would you like any further clarification or details?

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Here are 5 related questions you might want to explore:

1. How do you find the angle between two vectors in 2D?
2. What is the geometric meaning of the dot product?
3. How can you use the cross product to find the angle between vectors in 3D?
4. What happens if the dot product of two vectors is zero?
5. Can vectors have an angle greater than \( \pi \)?

**Tip:** The dot product can also be used to determine whether two vectors are perpendicular; if their dot product is zero, the vectors are orthogonal.

This solution explains how to find the angle between the vectors (9, 0, -9) and (9, 9, 0) using the dot product formula. Key concepts include vector geometry, trigonometry, and the cosine rule for determining angles between vectors. Suitable for Grades 10-12.

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