Problem 11 - Precalculus : Building Concepts and Connections

Question

Find the smallest nomnegation angle betwoen the vectors \(\mathbf{v}\) and \(\mathbf{w} .\) Round your anstoer to the nearest tenth of a degree. $$\mathbf{v}=(-2,0), \mathbf{w}=\langle 0,3\rangle$$

Step-by-Step Solution

Answer

The smallest nonnegation angle between the vectors is \(90^\circ\).

Step 1: Find the Dot Product

The dot product of vectors \(\mathbf{v} = (-2,0)\) and \(\mathbf{w} = (0,3)\) can be calculated as: \(\mathbf{v} \cdot \mathbf{w} = (-2*0 + 0*3) = 0\)

Step 2: Find the Magnitudes

The magnitude of vector \( \mathbf{v} \) is \( ||\mathbf{v}|| = \sqrt{-2^2 + 0^2} = 2 \) and the magnitude of vector \( \mathbf{w} \) is \( ||\mathbf{w}|| = \sqrt{0^2 + 3^2} = 3 \)

Step 3: Calculate the Angle

Using the formula for the angle between two vectors, cos(\( \theta \)) = \( \frac{\mathbf{v} \cdot \mathbf{w}}{||\mathbf{v}|| ||\mathbf{w}||} \), we get cos(\( \theta \)) = \( \frac{0}{2*3} = 0 \). So, \( \theta = cos^{-1}(0) = 90^\circ \).

Key Concepts

Dot Product

The dot product, also known as the scalar product, plays a pivotal role in finding the angle between two vectors. It is a way of multiplying two vectors to get a single scalar or real number. The dot product of two vectors \( \mathbf{u} \text{ and } \mathbf{v} \) with components \( \mathbf{u} = (u_1, u_2) \text{ and } \mathbf{v} = (v_1, v_2) \) is calculated as \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 \).

The result of the dot product tells us about the relationship between the two vectors. For instance, if the dot product is zero, as in the exercise with vectors \( \mathbf{v} \) and \( \mathbf{w} \), this indicates that the vectors are orthogonal, meaning they are at a right angle to each other. If the dot product is positive, the vectors make an acute angle; if it's negative, they form an obtuse angle.

This concept is very useful as it provides a means to calculate the angle between vectors using the cosine of the angle, which leads us to another important concept: cosine similarity.

Magnitude of a Vector

Understanding the magnitude of a vector is crucial when working with vectors and geometry. The magnitude refers to the length or size of the vector and is often denoted as \( ||\mathbf{v}|| \). For a 2D vector \( \mathbf{v} = (v_1, v_2) \), the magnitude is given by the square root of the sum of the squares of its components: \( ||\mathbf{v}|| = \sqrt{v_1^2 + v_2^2} \).

The concept of magnitude is comparable to measuring the straight-line distance from the origin to the point denoted by the vector in a coordinate system. For our exercise, the magnitudes of vectors \( \mathbf{v} \) and \( \mathbf{w} \) are the distances from the origin to the points (-2,0) and (0,3), respectively. Knowing the magnitudes is necessary to apply the formula for finding the angle between vectors using the dot product and cosine similarity.

Cosine Similarity

Cosine similarity is a measure that calculates the cosine of the angle between two non-zero vectors in a multi-dimensional space. It provides an indication of how similar the directional movements of the vectors are. The cosine similarity is obtained using the dot product and the magnitudes of the vectors and is defined by the formula \( \frac{\mathbf{u} \cdot \mathbf{v}}{||\mathbf{u}|| ||\mathbf{v}||} \), where \( \mathbf{u} \) and \( \mathbf{v} \) are the vectors in question.

In the context of our exercise, we used cosine similarity to find the angle between vectors \( \mathbf{v} \) and \( \mathbf{w} \). Since the dot product was zero, the cosine similarity also equaled zero, leading us to conclude that the angle between them is \( 90^\circ \), as the cosine of \( 90^\circ \) is zero. This measure is highly valuable in many fields, including machine learning, where it's used to assess the similarity between documents or vectors of features.