Problem 16 - 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. $$\mathbf{v}=\left\langle-2, \frac{3}{2}\right\rangle, \mathbf{w}=\langle 1,2\rangle$$

Step-by-Step Solution

Answer

The smallest nonnegative angle between the vectors \(\mathbf{v}\) and \(\mathbf{w}\) is approximately \(40.9\) degrees when rounded to the nearest tenth.

Step 1: Calculate the dot product

The dot product of \(\mathbf{v}\) and \(\mathbf{w}\) is obtained by multiplying corresponding components in each vector and summing the results: \((-2)(1) + (\frac{3}{2})(2) = -2 + 3 = 1\).

Step 2: Compute the magnitude of each vector

The magnitude of vector \(\mathbf{v}\) is: \(\sqrt{(-2)^2 + (\frac{3}{2})^2} = \sqrt{4 + \frac{9}{4}} = \sqrt{\frac{16}{4} + \frac{9}{4}} = \sqrt{\frac{25}{4}} = \frac{5}{2}\). The magnitude of vector \(\mathbf{w}\) is similarly computed: \(\sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}\).

Step 3: Calculate the angle

The angle θ is found using the equation \(\cos(\theta) = \frac{\mathbf{v}\cdot\mathbf{w}}{|\mathbf{v}| |\mathbf{w}|}\), where \(\mathbf{v}\cdot\mathbf{w}\) is the dot product, and \(|\mathbf{v}|\) and \(|\mathbf{w}|\) are the magnitudes of vectors \(\mathbf{v}\) and \(\mathbf{w}\) respectively. Substituting yields \(\cos(\theta) = \frac{1}{(\frac{5}{2})(\sqrt{5})}\). Thus, \(\theta = \cos^{-1}\left(\frac{1}{(\frac{5}{2})(\sqrt{5})}\right)\). Ensure to round this to the nearest tenth of a degree as initially stated.

Key Concepts

Dot Product

The dot product is a crucial concept in vector mathematics. It helps in understanding how two vectors interact in terms of their direction and magnitude. To compute the dot product of two vectors, like \(\mathbf{v}\) and \(\mathbf{w}\), you multiply their corresponding components and then add the results. For instance, if \(\mathbf{v} = \langle a, b \rangle\) and \(\mathbf{w} = \langle c, d \rangle\), the dot product is given by \(a \cdot c + b \cdot d\).

In our example, we calculated the dot product of \(\left\langle -2, \frac{3}{2} \right\rangle\) and \(\langle 1, 2 \rangle\):

\[-2 \times 1 + \left(\frac{3}{2}\right) \times 2 = -2 + 3 = 1\]

This result not only provides information for calculating vector angles but also indicates that the vectors are neither orthogonal (perpendicular) nor parallel, as the dot product is neither zero nor at its maximum value.

Vector Magnitude

Vector magnitude measures a vector's length in space, and it helps us understand the vector's size independently of direction. To find the magnitude of a vector \(\mathbf{v} = \langle a, b \rangle\) we use the formula:

\[|\mathbf{v}| = \sqrt{a^2 + b^2}\]

This approach is similar to calculating the distance between the origin and a point in a Cartesian plane.

Let’s look at our vectors from the example:

For \(\mathbf{v} = \left\langle -2, \frac{3}{2} \right\rangle\):

\(|\mathbf{v}| = \sqrt{(-2)^2 + \left(\frac{3}{2}\right)^2} = \sqrt{4 + \frac{9}{4}} = \frac{5}{2}\)

Now, for \(\mathbf{w} = \langle 1, 2 \rangle\):

\(|\mathbf{w}| = \sqrt{1^2 + 2^2} = \sqrt{5}\)

With these magnitudes, we can proceed to calculate the angle between the vectors using the dot product.

Cosine of Angle

The cosine of the angle between two vectors is an important measure used to determine the direction relationship between them. By calculating the cosine of the angle, we gain insight into whether vectors are pointing in similar or opposing directions.

The cosine of the angle \(\theta\) between two vectors \(\mathbf{v}\) and \(\mathbf{w}\) can be found using the formula:

\[\cos(\theta) = \frac{\mathbf{v} \cdot \mathbf{w}}{|\mathbf{v}| \cdot |\mathbf{w}|}\]

This requires the dot product and the magnitudes (lengths) of the vectors.

For example, in our exercise:

Calculating \(\cos(\theta)\) we have:

\[\cos(\theta) = \frac{1}{\left(\frac{5}{2}\right) \cdot \sqrt{5}}\]

Once you have this value, using an arccosine function or a calculator will allow you to find \(\theta\), which is the angle in degrees. Don't forget to round it to the nearest tenth. Understanding this formula is essential for any vector-related tasks, as it indicates how vectors line up with each other in space.