Problem 66 - Precalculus with Calculus Previews

Question

Find the angle between the given pair of vectors. Round your answer to two decimal places. $$ \langle 3,5\rangle,\langle-4,-2\rangle $$

Step-by-Step Solution

Answer

The angle between the vectors is approximately 147.90 degrees.

Step 1: Calculate the Dot Product

To find the angle between two vectors, we first calculate the dot product. The dot product formula for vectors \( \langle a, b \rangle \) and \( \langle c, d \rangle \) is given by: \( a \cdot c + b \cdot d \). For our vectors, this becomes: \( 3 \cdot (-4) + 5 \cdot (-2) = -12 - 10 = -22 \).

Step 2: Find the Magnitudes of Each Vector

Next, find the magnitude of each vector using the formula: \( \sqrt{a^2 + b^2} \).For \( \langle 3,5 \rangle \), the magnitude is: \( \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34} \).For \( \langle -4,-2 \rangle \), the magnitude is: \( \sqrt{(-4)^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} \).

Step 3: Calculate the Cosine of the Angle

The cosine of the angle \( \theta \) between two vectors is given by the formula: \( \cos(\theta) = \frac{\text{dot product}}{\text{magnitude of the first vector} \times \text{magnitude of the second vector}} \). Substituting the values we have: \( \cos(\theta) = \frac{-22}{\sqrt{34} \times \sqrt{20}} \approx \frac{-22}{\sqrt{680}} = \frac{-22}{26.077} \approx -0.843 \).

Step 4: Compute the Angle using Inverse Cosine

Finally, to find the angle \( \theta \) in degrees, use the inverse cosine function: \( \theta = \cos^{-1}(-0.843) \). Using a calculator, we find: \( \theta \approx 147.90 \) degrees.

Key Concepts

Dot Product

The dot product is a fundamental operation in vector mathematics. When you have two vectors, the dot product is calculated by multiplying corresponding components of these vectors and then summing those products. For the vectors \( \langle 3, 5 \rangle \) and \( \langle -4, -2 \rangle \), the dot product is computed as follows:

• Multiply the first components: \( 3 \cdot (-4) = -12 \).
• Multiply the second components: \( 5 \cdot (-2) = -10 \).

Add these results together to get the dot product: \(-12 + (-10) = -22 \).
This operation reveals how much one vector "acts" in the direction of another, considering their scaling values.
In our example, a negative dot product indicates that the vectors form an obtuse angle.

Vector Magnitudes

The magnitude of a vector measures its length. It's akin to finding the distance of a point from the origin in a 2D space. To compute the magnitude of a vector \( \langle a, b \rangle \), you use the formula:
\[ \text{Magnitude} = \sqrt{a^2 + b^2} \] For the vector \( \langle 3, 5 \rangle \), calculate its magnitude as:

• Square each component: \( 3^2 = 9 \) and \( 5^2 = 25 \).
• Add the squared components: \( 9 + 25 = 34 \).
• Take the square root: \( \sqrt{34} \).

Similarly, for \( \langle -4, -2 \rangle \), the magnitude is computed as:

• \( (-4)^2 = 16 \) and \( (-2)^2 = 4 \).
• Add these: \( 16 + 4 = 20 \).
• Square root: \( \sqrt{20} \).

These magnitudes give us the lengths of the vectors, essential for the next calculations.

Inverse Cosine Function

The inverse cosine function, denoted by \( \cos^{-1} \), helps us find the angle between two vectors once we have determined the cosine of that angle. The cosine \( \cos(\theta) \) can be found using the dot product and magnitudes' result:\[\cos(\theta) = \frac{\text{dot product}}{\text{magnitude of the first vector} \times \text{magnitude of the second vector}}\] For our vectors, substitute the values we calculated:

• \( \frac{-22}{\sqrt{34} \times \sqrt{20}} \approx \frac{-22}{26.077} \approx -0.843 \).

Now to find \( \theta \), use the inverse cosine:\( \theta = \cos^{-1}(-0.843) \).
Using a calculator, you find \( \theta \approx 147.90 \) degrees.
This step reveals the degree measure between the two vectors, completing the vector analysis.