Problem 5 - Contemporary Precalculus

Question

Find \(\boldsymbol{u} \cdot \boldsymbol{v}, \boldsymbol{u} \cdot \boldsymbol{u},\) and \(\boldsymbol{v} \cdot \boldsymbol{v}\) $$\mathbf{u}=3 \mathbf{i}+2 \mathbf{j}, \mathbf{v}=2 \mathbf{i}+3 \mathbf{j}$$

Step-by-Step Solution

Answer

Answer: The dot products for the given vectors are: 1. The dot product of vectors u and v is 12 ($$\mathbf{u} \cdot \mathbf{v} = 12$$). 2. The dot product of vector u and itself is 13 ($$\mathbf{u} \cdot \mathbf{u} = 13$$). 3. The dot product of vector v and itself is 13 ($$\mathbf{v} \cdot \mathbf{v} = 13$$).

Step 1: Identify the components of the vectors

Given vectors are: $$\mathbf{u} = 3\mathbf{i} + 2\mathbf{j}$$ $$\mathbf{v} = 2\mathbf{i} + 3\mathbf{j}$$ The components of the vectors are: $$\mathbf{u} = \langle 3, 2 \rangle$$ $$\mathbf{v} = \langle 2, 3 \rangle$$

Step 2: Find the dot product of u and v

To find the dot product of vectors u and v, we use the formula: $$\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2$$ Plugging in the components of vectors u and v: $$\mathbf{u} \cdot \mathbf{v} = (3)(2) + (2)(3) = 6 + 6 = 12$$

Step 3: Find the dot product of u and itself

To find the dot product of vector u and itself, use the formula: $$\mathbf{u} \cdot \mathbf{u} = u_1u_1 + u_2u_2$$ Plugging in the components of vector u: $$\mathbf{u} \cdot \mathbf{u} = (3)(3) + (2)(2) = 9 + 4 = 13$$

Step 4: Find the dot product of v and itself

To find the dot product of vector v and itself, use the formula: $$\mathbf{v} \cdot \mathbf{v} = v_1v_1 + v_2v_2$$ Plugging in the components of vector v: $$\mathbf{v} \cdot \mathbf{v} = (2)(2) + (3)(3) = 4 + 9 = 13$$ So, $$\mathbf{u} \cdot \mathbf{v} = 12$$, $$\mathbf{u} \cdot \mathbf{u} = 13$$, and $$\mathbf{v} \cdot \mathbf{v} = 13$$.

Key Concepts

Vector Components

Vector components are the building blocks of a vector that define its magnitude and direction in space. When working with two-dimensional vectors, like \( \mathbf{u} = 3\mathbf{i} + 2\mathbf{j} \), the vector is expressed in terms of its components along the horizontal (\( \mathbf{i} \)) and vertical (\( \mathbf{j} \)) axes. Here \( \mathbf{i} \) and \( \mathbf{j} \) are the unit vectors in the direction of the x-axis and y-axis, respectively.
Understanding the components is crucial because they allow us to calculate other important vector operations like adding vectors, finding magnitudes, and especially the dot product. For example, both vectors \( \mathbf{u} = \langle 3, 2 \rangle \) and \( \mathbf{v} = \langle 2, 3 \rangle \) can be described by their components \( 3 \) and \( 2 \) for \( \mathbf{u} \), and \( 2 \) and \( 3 \) for \( \mathbf{v} \). These components are simply the coefficients of \( \mathbf{i} \) and \( \mathbf{j} \).
Once you have identified the components, you lay the groundwork for further vector operations through vector algebra.

Vector Algebra

Vector algebra involves performing mathematical operations on vectors. Unlike numbers, vectors have both magnitude and direction, making them unique. The basic operations include addition, subtraction, and multiplication of vectors.
When multiplying vectors, you'll commonly deal with the dot product and the cross product. In this context, we focus on the dot product, which is a scalar value outcome rather than a vector. It's calculated using the formula:

• For vectors \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \), the dot product is \( \mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 \).

This operation gives us insights into the relation between the vectors, specifically indicating how much of one vector goes in the direction of another. A dot product of zero indicates that the vectors are orthogonal, or perpendicular to each other.

Vector Multiplication

Vector multiplication can refer to several operations, but here we focus on the dot product, an operation of deterministic significance. Given vectors \( \mathbf{u} \) and \( \mathbf{v} \), the dot product is a measure of how aligned these vectors are. Its formula is:

• \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 \).

Using the vectors from our example, \( \mathbf{u} = \langle 3, 2 \rangle \) and \( \mathbf{v} = \langle 2, 3 \rangle \), we find \( \mathbf{u} \cdot \mathbf{v} = 3\times2 + 2\times3 = 12 \).
This scalar result tells us about the geometric relationship between \( \mathbf{u} \) and \( \mathbf{v} \). If it's positive, like it is here, it means the angle between them is less than 90°, indicating some degree of alignment.
Calculating the dot product of a vector with itself, \( \mathbf{u} \cdot \mathbf{u} \), involves squaring its components and summing them, effectively providing the square of the vector's magnitude.