Problem 67 - Algebra and Trigonometry

Question

Describe how to find the angle between two vectors.

Step-by-Step Solution

Answer

To find the angle between two vectors, we calculate their dot product and the magnitudes of the individual vectors. We then divide the dot product by the product of their magnitudes and find the inverse cosine of the result to get the angle.

Step 1: Understand the Dot Product

Recall the definition of the dot product of two vectors. If \(\vec{a} = [a_1, a_2, ..., a_n]\) and \(\vec{b} = [b_1, b_2, ..., b_n]\), the dot product is given as \( \vec{a} . \vec{b} = a_1 * b_1 + a_2 * b_2 + ... + a_n * b_n \).

Step 2: Explore the Angle between Two Vectors

The angle between two vectors, say \(\vec{a}\) and \(\vec{b}\), can be calculated using their dot product and the magnitudes of the vectors by the formula: \( cos(\Theta) = (\vec{a}.\vec{b})/ (\|\vec{a}\| * \|\vec{b}\| ) \), where \(\|\vec{a}\|\) denotes the magnitude or length of vector \(\vec{a}\), \(\|\vec{b}\|\) denotes the magnitude or length of vector \(\vec{b}\), and \(\Theta\) is the angle between \(\vec{a}\) and \(\vec{b}\).

Step 3: Finding Magnitudes of Vectors

Calculate magnitudes of vectors using the formula \(\|\vec{a}\| = \sqrt{a_1^2 + a_2^2 + ... + a_n^2}\) for vector \(\vec{a}\) and similarly for vector \(\vec{b}\).

Step 4: Implementing the Formula

Substitute the values of the dot product and the magnitudes of the vectors into the formula to find the value of \(cos(\Theta)\). The angle \(\Theta\) can be found using the inverse cosine or arccos function.