Problem 13 - Precalculus

Question

Find the angle (round to the nearest degree) between each pair of vectors. $$\langle-4,3\rangle \text { and }\langle-5,-9\rangle$$

Step-by-Step Solution

Answer

The angle between the vectors is approximately 98°.

Step 1: Recall the Dot Product Formula

The formula to find the dot product between two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \) is \( \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 \).

Step 2: Compute the Dot Product

Substitute the vectors into the dot product formula: \( \langle -4, 3 \rangle \cdot \langle -5, -9 \rangle = (-4)(-5) + (3)(-9) = 20 - 27 = -7 \).

Step 3: Recall the Magnitude Formula

The magnitude of a vector \( \mathbf{a} = \langle a_1, a_2 \rangle \) is \( \| \mathbf{a} \| = \sqrt{a_1^2 + a_2^2} \).

Step 4: Calculate Magnitudes of Vectors

Calculate magnitude of \( \langle -4, 3 \rangle \): \( \sqrt{(-4)^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \). Calculate magnitude of \( \langle -5, -9 \rangle \): \( \sqrt{(-5)^2 + (-9)^2} = \sqrt{25 + 81} = \sqrt{106} \approx 10.2956 \).

Step 5: Use the Dot Product to Find the Cosine of the Angle

The cosine of the angle \( \theta \) between two vectors is \( \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\| \mathbf{a} \| \| \mathbf{b} \|} \). Substitute values: \( \cos \theta = \frac{-7}{5 \times 10.2956} = \frac{-7}{51.478} \approx -0.136 \).

Step 6: Calculate the Angle

Find \( \theta \) by taking the inverse cosine: \( \theta = \cos^{-1}(-0.136) \). Use a calculator to find \( \theta \approx 97.8^\circ \).

Step 7: Round to the Nearest Degree

Round \( 97.8^\circ \) to the nearest degree. The angle between the vectors is approximately \( 98^\circ \).

Key Concepts

Dot Product

When working with vectors, the dot product, often referred to as the scalar product, is a key operation used to find the angle between them. This operation combines the two vectors into a single scalar or real number.
It is calculated by multiplying the corresponding components of the vectors and then summing those products.Here's a simple way to remember it:

• For vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \), the dot product is: \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \).

This formula can be extended to any number of dimensions, but our focus is on two-dimensional vectors. Learning to compute the dot product is a fundamental step in linear algebra and vector geometry,
as it serves as the bridge to calculating the angle between the vectors.

Magnitude of a Vector

The magnitude of a vector is essentially its "length" or "size" in the plane. It tells us how long the vector is without considering the direction.To calculate the magnitude of a vector \( \mathbf{a} = \langle a_1, a_2 \rangle \), use this formula:

• \( \| \mathbf{a} \| = \sqrt{a_1^2 + a_2^2} \)

This formula applies the Pythagorean theorem, where the components of the vector represent the two legs of a right triangle,
and the magnitude is the hypotenuse. Understanding the magnitude is essential
because it allows you to scale the vector and relates to other calculations, such as finding the unit vector.

Cosine of an Angle

Finally, finding the cosine of an angle between two vectors involves using both the dot product and the magnitudes of the vectors.
This is important because the cosine gives us a way to determine how "aligned" two vectors are.The formula for the cosine of the angle \( \theta \) between the vectors \( \mathbf{a} \) and \( \mathbf{b} \) is:

• \( \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\| \mathbf{a} \| \| \mathbf{b} \|} \)

Using this formula, you'll determine if vectors are parallel, orthogonal, or neither. If \( \cos \theta = 1 \) or \( -1 \), the vectors are parallel but in opposite directions,
respectively. If \( \cos \theta = 0 \), the vectors are orthogonal, or perpendicular.
Calculating the cosine is usually the final step in determining the precise angle as it allows us to use inverse trigonometric functions
to find the angle itself.