Problem 41 - Precalculus with Limits

Question

In Exercises 41-44, graph the vectors and find the degree measure of the angle \(\theta\) between the vectors. \(\mathbf{u} = 3\mathbf{i} + 4\mathbf{j}\) \(\mathbf{v} = -7\mathbf{i} + 5\mathbf{j}\)

Step-by-Step Solution

Answer

The degree measure of the angle between vectors \(\mathbf{u}\) and \(\mathbf{v}\) is obtained by evaluating \(\theta = \text{cos}^{-1} \left( \frac{-1}{5*\sqrt{74}} \right)\).

Step 1: Representing Vectors

Vectors \(\mathbf{u}\) and \(\mathbf{v}\) can be represented graphically in the cartesian coordinate system with \(\mathbf{i}\) and \(\mathbf{j}\) components corresponding to x and y direction respectively. The vector \(\mathbf{u}\) represented by the coordinates (3,4) and vector \(\mathbf{v}\) represented by coordinates (-7,5). Draw those points in the 2D plane along with the vectors from origin to these points.

Step 2: Calculating Dot Product

The dot product of the vectors is calculated as \(\mathbf{u} . \mathbf{v} = 3*(-7) + 4*5 = -21 + 20 = -1.\)

Step 3: Calculating Magnitudes

The magnitudes of the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are calculated as \(\|\mathbf{u}\| = \sqrt{3^2 + 4^2} = 5,\) and \(\|\mathbf{v}\| = \sqrt{(-7)^2 + 5^2} = \sqrt{74}\) respectively.

Step 4: Find Angle Between Vectors

The angle between the vectors is given by \(\theta = \text{cos}^{-1} \left( \frac{\mathbf{u} . \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|} \right)\). Substituting the values from previous steps, we find \(\theta = \text{cos}^{-1} \left( \frac{-1}{5*\sqrt{74}} \right)\) which yields the degree measure of the angle \(\theta\).

Key Concepts

Dot Product

The dot product is a way to multiply two vectors. It's a scalar quantity, meaning it doesn't have a direction, only a magnitude. When you calculate the dot product of two vectors, you are essentially measuring how much one vector goes in the direction of the other. This is particularly useful in finding the angle between vectors.
To calculate the dot product of two vectors in a 2D space, say \( \mathbf{u} = a\mathbf{i} + b\mathbf{j} \) and \( \mathbf{v} = c\mathbf{i} + d\mathbf{j} \), use this formula:

• \( \mathbf{u} \cdot \mathbf{v} = a \cdot c + b \cdot d \)

The components are multiplied together and then added. In our example, the dot product is \( 3(-7) + 4(5) = -21 + 20 = -1 \).
This negative result tells us that the angle between \( \mathbf{u} \) and \( \mathbf{v} \) is greater than 90 degrees, indicating that they point more away from each other than towards each other.

Magnitude of Vectors

The magnitude (or length) of a vector helps describe how long, or large, the vector is. To find the magnitude, think of it as applying the Pythagorean theorem to the vector's components.
For a vector \( \mathbf{u} = a\mathbf{i} + b\mathbf{j} \), the magnitude is calculated using:

• \( \|\mathbf{u}\| = \sqrt{a^2 + b^2} \)

This gives you the 'length' of the vector from the origin to its point in space.
In our scenario, for \( \mathbf{u} = 3\mathbf{i} + 4\mathbf{j} \), the magnitude is \( \sqrt{3^2 + 4^2} = 5 \). And for \( \mathbf{v} = -7\mathbf{i} + 5\mathbf{j} \), it is \( \sqrt{(-7)^2 + 5^2} = \sqrt{74} \). These magnitudes help determine how far each vector stretches from the origin.

Angle Between Vectors

The angle between vectors is an important quantity in mathematics and physics, as it tells us how the vectors are oriented with respect to each other.
To find this angle, \( \theta \), between two vectors \( \mathbf{u} \) and \( \mathbf{v} \), we use the formula:

• \( \theta = \cos^{-1} \left( \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|} \right) \)

The angle is derived from the dot product and the magnitudes of the two vectors. In essence, this formula is using the properties of cosine to express the relation between the vectors.
For the vectors given, the calculation becomes \( \theta = \cos^{-1} \left( \frac{-1}{5 \times \sqrt{74}} \right) \). By resolving this, you find the angle in degrees, indicating how closely aligned or opposed the vectors are.