Problem 34 - Precalculus with Limits

Question

In Exercises 31-40, find the angle \(\theta\) between the vectors. \(\mathbf{u} = 2\mathbf{i} - 3\mathbf{j}\) \(\mathbf{v} = \mathbf{i} - 2\mathbf{j}\)

Step-by-Step Solution

Answer

The angle \( \theta \) between the vectors \( \mathbf{u} \) and \( \mathbf{v} \) is approximately 0.32 radians or 18.43 degrees.

Step 1: Compute the Dot Product of \(\mathbf{u}\) and \(\mathbf{v}\)

The dot product of two vectors \( \mathbf{u} \) and \( \mathbf{v} \) is calculated by multiplying corresponding elements and summing the results: \( \mathbf{u} \cdot \mathbf{v} = (2)(1) - (3)(-2) = 2 + 6 = 8

Step 2: Compute the Magnitude of \(\mathbf{u}\) and \(\mathbf{v}\)

The magnitude (or length) of a vector can be calculated using the Pythagorean theorem: \( ||\mathbf{u}|| = \sqrt{ (2)^2 + (-3)^2 } = \sqrt{4 + 9} = \sqrt{13} \) and \( ||\mathbf{v}|| = \sqrt{ (1)^2 + (-2)^2 } = \sqrt{1 + 4} = \sqrt{5} \).

Step 3: Find the Angle Between \(\mathbf{u}\) and \(\mathbf{v}\)

Substitute the dot product and the magnitudes into the formula for the angle \( \theta \): \( \theta = \arccos \left ( \frac{\mathbf{u} \cdot \mathbf{v}}{||\mathbf{u}|| ||\mathbf{v}||} \right ) = \arccos \left ( \frac{8}{\sqrt{13} \cdot \sqrt{5}} \right ) \). This will give the angle in radians. To convert it to degrees, use that \( \pi \) radians is equal to 180 degrees.

Key Concepts

Dot Product

To grasp the concept of the dot product, think of it as a way to multiply two vectors to get a single number, rather than another vector. The dot product essentially measures how much one vector "goes in the direction" of another. The formula for the dot product of two vectors, \( \mathbf{u} = a\mathbf{i} + b\mathbf{j} \) and \( \mathbf{v} = c\mathbf{i} + d\mathbf{j} \), is:

• \( \mathbf{u} \cdot \mathbf{v} = ac + bd \)

In our case, \( \mathbf{u} = 2\mathbf{i} - 3\mathbf{j} \) and \( \mathbf{v} = \mathbf{i} - 2\mathbf{j} \). Calculating their dot product involves:- \( 2 \times 1 = 2 \)- \( -3 \times -2 = 6 \)Summing these, we obtain a dot product of 8. The dot product can be positive, negative, or zero; here, it is positive, indicating vectors generally point in a similar direction.
The dot product is crucial in determining angles between vectors, providing a bridge from vector components to geometric properties.

Magnitude of a Vector

The magnitude, or length, of a vector is like its "size." It is calculated using the Pythagorean theorem, giving a sense of how long the vector is in two-dimensional space. The formula for finding the magnitude of a vector \( \mathbf{u} = a\mathbf{i} + b\mathbf{j} \) is:

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

For vector \( \mathbf{u} = 2\mathbf{i} - 3\mathbf{j} \), the magnitude is:- \( \sqrt{(2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \)
For vector \( \mathbf{v} = \mathbf{i} - 2\mathbf{j} \), the magnitude is:- \( \sqrt{(1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \)
Understanding vector magnitude helps in finding the unit vector, which is another vector that points in the same direction but with a length of 1. You'll find it useful in many applications such as physics and engineering.

Vector Arithmetic

Vector arithmetic involves operations such as addition, subtraction, and scalar multiplication. These simple yet foundational actions allow us to manipulate vectors in various ways.When adding or subtracting two vectors, each component of one vector is added to or subtracted from the corresponding component of the other. For example, if you have vectors \( \mathbf{u} = a\mathbf{i} + b\mathbf{j} \) and \( \mathbf{v} = c\mathbf{i} + d\mathbf{j} \), then:

• \( \mathbf{u} + \mathbf{v} = (a + c)\mathbf{i} + (b + d)\mathbf{j} \)
• \( \mathbf{u} - \mathbf{v} = (a - c)\mathbf{i} + (b - d)\mathbf{j} \)

Scalar multiplication involves multiplying each component of a vector by a scalar, or number:- If \( k \) is a scalar, then \( k\mathbf{u} = ka\mathbf{i} + kb\mathbf{j} \)These operations are essential building blocks in vector calculus and physics, where they form the basis of more complex computations like finding resultant vectors, which combine vectors in magnitude and direction.