Problem 2 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-8: Find (a) the dot product of the two vectors and (b) the angle between the two vectors. $$ \langle 4,-7\rangle, \quad\langle-2,3\rangle $$

Step-by-Step Solution

Answer

The dot product is -29, and the angle is \( \cos^{-1}\left(\frac{-29}{\sqrt{845}}\right)\).

Step 1: Calculate the Dot Product

To find the dot product of two vectors \( \langle a,b \rangle \) and \( \langle c,d \rangle \), use the formula: \[ a \times c + b \times d \]. For the vectors \( \langle 4,-7 \rangle \) and \( \langle -2, 3 \rangle \), compute the dot product: \( 4 \times -2 + (-7) \times 3 \). This simplifies to \(-8 - 21 = -29 \). The dot product is \(-29\).

Step 2: Calculate the Magnitude of Each Vector

To calculate the magnitude of a vector \( \langle a, b \rangle \), use the formula: \[ \sqrt{a^2 + b^2} \]. Calculate the magnitude of \( \langle 4,-7 \rangle \): \( \sqrt{4^2 + (-7)^2} = \sqrt{16 + 49} = \sqrt{65} \). Calculate the magnitude of \( \langle -2,3 \rangle \): \( \sqrt{(-2)^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \).

Step 3: Calculate the Angle Between the Vectors

The formula to find the angle \( \theta \) between two vectors is: \[ \cos \theta = \frac{\text{dot product}}{\text{magnitude of first vector} \times \text{magnitude of second vector}} \].Substitute the known values: \( \cos \theta = \frac{-29}{\sqrt{65} \times \sqrt{13}} \).This simplifies to \( \cos \theta = \frac{-29}{\sqrt{845}} \).Therefore, \( \theta = \cos^{-1}\left(\frac{-29}{\sqrt{845}}\right)\). Calculate this using a calculator to find the angle.

Key Concepts

Vector Calculations

Vectors are fundamental in physics and engineering. They help describe quantities with both magnitude and direction. When working with vectors, one of the essential tasks is calculating the dot product. The dot product is an algebraic operation that combines two vectors, producing a scalar (a single number).

• Formula: To find the dot product of two vectors, \( \langle a, b \rangle \) and \( \langle c, d \rangle \), use the formula: \( a \times c + b \times d \).
• Example: For the vectors \( \langle 4, -7 \rangle \) and \( \langle -2, 3 \rangle \), you calculate it as follows: \( 4 \times -2 + (-7) \times 3 = -8 - 21 = -29 \).

Understanding the dot product is crucial because it not only produces a scalar but also provides valuable information about the relationship between the two vectors. A dot product of zero, for example, indicates that the vectors are orthogonal, meaning they are at a right angle to each other.

Magnitude Calculation

The magnitude of a vector measures its length in space. It helps you understand how "strong" or "intense" a vector is, irrespective of its direction. Calculating the magnitude is essential for normalizing vectors or computing other quantities such as work or energy in physics.

• Formula: The magnitude of a vector \( \langle a, b \rangle \) is given by \( \sqrt{a^2 + b^2} \).
• Example: For the vector \( \langle 4, -7 \rangle \), the magnitude is \( \sqrt{4^2 + (-7)^2} = \sqrt{16 + 49} = \sqrt{65} \).
• For the vector \( \langle -2, 3 \rangle \), the magnitude is \( \sqrt{(-2)^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \).

Using magnitudes, we can also understand the concept of normalization, where a vector is scaled to have a magnitude of 1, keeping the direction the same.

Angle Between Vectors

Finding the angle between two vectors involves combining concepts of dot product and magnitude. This angle indicates how aligned or clashing two vectors are. It is particularly useful in physics and engineering when examining forces or motions.

• Formula: The angle \( \theta \) between two vectors can be found using \( \cos \theta = \frac{\text{dot product}}{\text{magnitude of first vector} \times \text{magnitude of second vector}} \).
• Example Calculation: Using the vectors from our example, \( \cos \theta = \frac{-29}{\sqrt{65} \times \sqrt{13}} \).
• This evaluates to \( \cos \theta = \frac{-29}{\sqrt{845}} \).
• Then, \( \theta = \cos^{-1}\left(\frac{-29}{\sqrt{845}}\right) \), which can be estimated using a calculator.

Calculating the angle reveals the directional difference between the vectors. If \( \theta \) is 90 degrees, they are perpendicular. If 0 degrees, they are parallel.