Problem 9 - 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. $$\mathbf{v}=\langle 1,3\rangle, \mathbf{w}=\langle-2,0\rangle$$

Step-by-Step Solution

Answer

The smallest nonnegative angle between the vectors is approximately 126.9 degrees.

Step 1: Calculate the Dot Product

The dot product of the vectors \(\mathbf{v}\) and \(\mathbf{w}\) is found by multiplying corresponding components and adding them. So, \(\mathbf{v}\cdot\mathbf{w} = (1*-2) + (3*0) = -2.\)

Step 2: Calculate the Magnitudes

The magnitude of a vector is found through the formula: \( ||\mathbf{v}|| = \sqrt{x^2 + y^2} \). So, \( ||\mathbf{v}|| = \sqrt{1^2 + 3^2} = \sqrt{10} \) and \( ||\mathbf{w}|| = \sqrt{(-2)^2 + 0^2} = 2.\)

Step 3: Use the Formula

Next, we use the formula to find the cosine of the angle, cos\( \theta \) = \(\mathbf{v}\cdot\mathbf{w}/||\mathbf{v}|| ||\mathbf{w}|| = -2/(2*\sqrt{10}) = -1/\sqrt{10}.\)

Step 4: Calculate the Angle

The final step is to calculate the angle in degrees. We take the inverse cosine (or arccosine) of the result from Step 3 and multiply by \(180/\pi\) to convert from radians to degrees. Therefore, \(\theta = \arccos(-1/\sqrt{10}) * 180/\pi \approx 126.9\) degrees.

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector mathematics. It allows us to combine two vectors to produce a single scalar, or number.
It provides useful information about the vectors, such as whether they are orthogonal (perpendicular) or aligned in a certain way. For two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \), the dot product is found using the formula:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \)

For our specific case with vectors \( \mathbf{v} = \langle 1, 3 \rangle \) and \( \mathbf{w} = \langle -2, 0 \rangle \), the dot productgave us \( \mathbf{v} \cdot \mathbf{w} = (1)(-2) + (3)(0) = -2 \).
This result can tell us about the angle between the vectors. If this product had been zero, the vectors would be perpendicular. However, a negative dot product suggests that the angle between the vectors is obtuse,meaning it's greater than 90 degrees.

Magnitude of a Vector

The magnitude of a vector is a measure of its length. Think of it like measuring distance; it's always a positive value or zero if the vector is a point. For a vector \( \mathbf{a} = \langle a_1, a_2 \rangle \), the magnitude is calculated as follows:

• \[ ||\mathbf{a}|| = \sqrt{a_1^2 + a_2^2} \]

When you calculate the magnitude of \( \mathbf{v} = \langle 1, 3 \rangle \), you find \( ||\mathbf{v}|| = \sqrt{1^2 + 3^2} = \sqrt{10} \).
Similarly, for \( \mathbf{w} = \langle -2, 0 \rangle \), we find \( ||\mathbf{w}|| = \sqrt{(-2)^2 + 0^2} = 2 \). Knowing the magnitudes of vectors is crucial when determining the angle between them, as these values are used in the cosine formula.

Angle Between Vectors

The angle between vectors is a key measurement in understanding how two vectors relate to each other in space. It can tell us whether vectors point in similar or opposite directions. To find this angle, we utilize the dot product along with the magnitudes of the vectors, using the formula:

• Cosine of the angle: \( \cos \theta = \frac{\mathbf{v} \cdot \mathbf{w}}{||\mathbf{v}|| \, ||\mathbf{w}||} \)

Using our vectors \( \mathbf{v} \) and \( \mathbf{w} \), we already calculated the dot product to be \(-2\), and we know the magnitudes are \( \sqrt{10} \) and \(2\), respectively.
Thus, the formula to find the cosine of the angle comes out to \( \cos \theta = \frac{-2}{2 \times \sqrt{10}} = \frac{-1}{\sqrt{10}} \).
To derive the angle in degrees, we must use the inverse cosine function \( \theta = \cos^{-1} \left( \frac{-1}{\sqrt{10}} \right) \).
Finally, this angle is converted from radians to degrees by multiplying with \( \frac{180}{\pi} \), resulting in approximately \( 126.9 \) degrees.
This angle corresponds with the earlier notion that the dot product was negative, indicating an obtuse angle of more than 90 degrees but less than 180 degrees.