To find the angle $\alpha$ between two vectors, we use the formula:

$\cos(\alpha) = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|}$

Where:

• $\mathbf{u}$ and $\mathbf{v}$ are the given vectors.
• $\mathbf{u} \cdot \mathbf{v}$ is the dot product of the two vectors.
• $\|\mathbf{u}\|$ and $\|\mathbf{v}\|$ are the magnitudes (norms) of the vectors.

For the given vectors: $\mathbf{u} = \begin{bmatrix} -3 \\ 2 \\ -3 \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} 4 \\ -3 \\ -1 \end{bmatrix}$

Step 1: Compute the dot product

$\mathbf{u} \cdot \mathbf{v} = (-3)(4) + (2)(-3) + (-3)(-1) = -12 - 6 + 3 = -15$

Step 2: Compute the magnitudes of $\mathbf{u}$ and $\mathbf{v}$

$\|\mathbf{u}\| = \sqrt{(-3)^2 + 2^2 + (-3)^2} = \sqrt{9 + 4 + 9} = \sqrt{22}$ $\|\mathbf{v}\| = \sqrt{(4)^2 + (-3)^2 + (-1)^2} = \sqrt{16 + 9 + 1} = \sqrt{26}$

Step 3: Compute $\cos(\alpha)$

$\cos(\alpha) = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|} = \frac{-15}{\sqrt{22} \cdot \sqrt{26}} = \frac{-15}{\sqrt{572}}$

Step 4: Compute $\alpha$ (angle in radians or degrees)

$\alpha = \cos^{-1}\left(\frac{-15}{\sqrt{572}}\right)$

Let me calculate this value explicitly. math

Given values

dot_product = -15 magnitude_u = math.sqrt(22) magnitude_v = math.sqrt(26)

Calculate cos(alpha)

cos_alpha = dot_product / (magnitude_u * magnitude_v)

Calculate the angle in radians

alpha_radians = math.acos(cos_alpha)

Convert to degrees

alpha_degrees = math.degrees(alpha_radians) alpha_radians, alpha_degreesThe angle $\alpha$ between the vectors is approximately:

• 2.25 radians
• 128.84 degrees

Would you like further clarification or more steps? Here are some related questions you might consider:

1. What is the formula for finding the angle between vectors in higher dimensions?
2. Can the angle between two vectors ever be negative?
3. How do orthogonal vectors relate to this calculation?
4. What are some applications of finding angles between vectors in physics or engineering?
5. How does the dot product determine the cosine of the angle?

Tip: Always double-check your vector components when calculating dot products to avoid sign errors!