Problem 23 - Precalculus

Question

Find the angle (round to the nearest degree) between each pair of vectors. $$\langle 4,6\rangle \text { and }\langle-6,-9\rangle$$

Step-by-Step Solution

Answer

The angle between the vectors is approximately 168 degrees.

Step 1: Determine the Dot Product

The dot product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) represented as \( \langle a_1, a_2 \rangle \) and \( \langle b_1, b_2 \rangle \), respectively, is calculated using the formula: \[\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2\]For the given vectors \( \langle 4, 6 \rangle \) and \( \langle -6, -9 \rangle \), the dot product is:\[4(-6) + 6(-9) = -24 - 54 = -78\]

Step 2: Calculate the Magnitude of Each Vector

The magnitude of a vector \( \mathbf{a} = \langle a_1, a_2 \rangle \) is calculated using the formula:\[\| \mathbf{a} \| = \sqrt{a_1^2 + a_2^2}\]For the vector \( \langle 4, 6 \rangle \):\[\| \langle 4, 6 \rangle \| = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52}\]For the vector \( \langle -6, -9 \rangle \):\[\| \langle -6, -9 \rangle \| = \sqrt{(-6)^2 + (-9)^2} = \sqrt{36 + 81} = \sqrt{117}\]

Step 3: Use the Dot Product Formula to Find the Cosine of the Angle

The cosine of the angle \( \theta \) between two vectors can be found using the formula:\[\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\| \mathbf{a} \| \cdot \| \mathbf{b} \|}\]Substituting the values we found:\[\cos \theta = \frac{-78}{\sqrt{52} \cdot \sqrt{117}}\]

Step 4: Calculate the Cosine and the Angle \( \theta \)

First, calculate the denominator:\[\sqrt{52} \cdot \sqrt{117} = \sqrt{6092}\]Then, find \( \cos \theta \):\[\cos \theta = \frac{-78}{\sqrt{6092}}\]Using a calculator, you find \( \cos \theta \approx -0.9879 \).

Step 5: Find the Angle in Degrees

Find the angle \( \theta \) in degrees by taking the inverse cosine (arccos) of \( \cos \theta \):\[\theta = \cos^{-1}(-0.9879)\]Using a calculator, \( \theta \approx 168.06 \) degrees.

Key Concepts

Dot Product

The dot product is a cornerstone concept in vector mathematics, frequently used to find angles between vectors. It simplifies to a single number, calculated from two vectors. For vectors represented as \( \langle a_1, a_2 \rangle \) and \( \langle b_1, b_2 \rangle \), the dot product formula is:

• Calculate the product of their corresponding components: \( a_1 \times b_1 \) and \( a_2 \times b_2 \).
• Add these results together: \( \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 \).

The purpose of the dot product is not only to get a result but to use it in combination with the magnitudes of the vectors to find the angle between them.
This result helps us understand how much one vector points in the direction of another.By understanding and applying the dot product formula, you can find the specific behavior of interacting vectors, such as alignment or orthogonality within geometric contexts.

Magnitude of a Vector

The magnitude of a vector is like the vector's length, representing how far the vector extends in space from its origin. To calculate it, use the formula for a vector \( \mathbf{a} = \langle a_1, a_2 \rangle \):

• Square each component of the vector: \( a_1^2 \) and \( a_2^2 \).
• Add the squared values: \( a_1^2 + a_2^2 \).
• Take the square root of this sum to get the magnitude: \( \| \mathbf{a} \| = \sqrt{a_1^2 + a_2^2} \).

Understanding vector magnitude is crucial because it normalizes a vector, helping to compare vectors regardless of their scale.
This concept is pivotal in the angle calculation between vectors, as it factors into determining how directional components contribute to their interaction.
By grasping this, you lay the groundwork for operating with vectors in physics and engineering scenarios.

Cosine of Angle Between Vectors

The cosine of the angle between two vectors is a vital concept that measures the relative orientation between them. Calculating \( \cos \theta \), where \( \theta \) is the angle, involves both the dot product and the magnitudes of the vectors:

• First, compute the dot product of the vectors \( \mathbf{a} \) and \( \mathbf{b} \).
• Find the magnitudes \( \| \mathbf{a} \| \) and \( \| \mathbf{b} \| \) of the two vectors.
• Divide the dot product by the product of these magnitudes: \( \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\| \mathbf{a} \| \cdot \| \mathbf{b} \|} \).

When \( \cos \theta \) is calculated, taking the inverse cosine will provide the angle between the vectors. This process unveils whether the vectors point in similar directions (positive cosine) or oppose each other (negative cosine).
Having a firm grasp on this concept can illuminate how physical phenomena such as forces or velocities interact directionally within a system.