Angle Between Vectors

Implement a function to calculate the angle between two vectors a and b in degrees. This task involves understanding vector operations and trigonometric relationships, particularly the dot product and its connection to the cosine of the angle between vectors.

The dot product of two vectors a and b is given by $\mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^{n} a_i b_i$ , where $n$ is the dimension of the vectors. The magnitude of a vector a is denoted by $||\mathbf{a}|| = \sqrt{\mathbf{a} \cdot \mathbf{a}}$ . To find the angle $\theta$ between two vectors, we use the formula $\cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{||\mathbf{a}|| \cdot ||\mathbf{b}||}$ .

Here are the steps to calculate the angle:

Compute the dot product of vectors a and b.
Calculate the magnitudes of vectors a and b.
Use the formula to find $\cos(\theta)$ .
Find the angle $\theta$ using the inverse cosine function.

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

This technique is widely used in computer vision for measuring similarity between feature vectors.

Example:

Input:

angle_between([1, 0], [0, 1])

Output:

90.00

Reasoning:

Step-by-step calculation using the angle formula:

$\theta = \arccos\left(\frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \times \|\mathbf{b}\|}\right)$

Compute dot product: $\mathbf{a} \cdot \mathbf{b} = (1 \times 0) + (0 \times 1) = 0$
Compute magnitudes: $\|\mathbf{a}\| = \sqrt{1^2 + 0^2} = 1$ $\|\mathbf{b}\| = \sqrt{0^2 + 1^2} = 1$
Apply the formula: $\cos(\theta) = \frac{0}{1 \times 1} = 0$
Find the angle: $\theta = \arccos(0) = 90°$
Result: $90.00°$

The vectors $[1,0]$ and $[0,1]$ are the standard basis vectors pointing along the x and y axes respectively, which are perpendicular ( $90°$ apart).

Constraints:

Both vectors have the same length n where 1 ≤ n ≤ 1000
Neither vector is the zero vector
Return the angle in degrees, rounded to 2 decimal places

Example:

Constraints:

Test Results