Problem 1 - Contemporary Precalculus

Question

Find \(\boldsymbol{u} \cdot \boldsymbol{v}, \boldsymbol{u} \cdot \boldsymbol{u},\) and \(\boldsymbol{v} \cdot \boldsymbol{v}\) $$\mathbf{u}=\langle 3,4\rangle, \mathbf{v}=\langle-5,2\rangle$$

Step-by-Step Solution

Answer

Answer: The scalar dot products are: - Vector u · Vector v = -7 - Vector u · Vector u = 25 - Vector v · Vector v = 29

Step 1: Find the dot product of u and v (u · v)

Recall the formula for computing the dot product between two vectors: u · v = (u_1 * v_1) + (u_2 * v_2) Plugging in the given values of u = 〈3, 4〉, and v = 〈-5, 2〉, we get: u · v = (3 * -5) + (4 * 2) = -15 + 8 = -7

Step 2: Find the dot product of u and u (u · u)

We need to find u · u using the same formula, but now the components of both vectors are the same as vector u: u · u = (u_1 * u_1) + (u_2 * u_2) u · u = (3 * 3) + (4 * 4) = 9 + 16 = 25

Step 3: Find the dot product of v and v (v · v)

Lastly, we need to compute v · v using the same formula but all components now belong to vector v: v · v = (v_1 * v_1) + (v_2 * v_2) v · v = (-5 * -5) + (2 * 2) = 25 + 4 = 29 In summary, these are the scalar dot products calculated: - Vector u · Vector v = -7 - Vector u · Vector u = 25 - Vector v · Vector v = 29

Key Concepts

Understanding Vectors

Vectors are fundamental entities in precalculus and mathematics that represent both a direction and a magnitude. They are used to describe movement, forces, and more in a geometrical and physical context. In the realm of a two-dimensional space, vectors are often represented as pairs of numbers, for example \( \langle 3, 4 \rangle \). This means the vector has a horizontal component of 3 and a vertical component of 4.
Working with vectors involves operations such as addition, subtraction, and importantly for this context, dot products. Understanding vectors paves the way for solving complex problems involving physics, engineering, and further mathematical studies. When working with vectors, always pay attention to their components, which dictate the vector's unique properties and applications.
In summary, vectors are crucial tools in representing quantities that have both size and direction, simplifying the study of various phenomena.

Dot Product in Precalculus

The dot product, also known as the scalar product, is a key operation you encounter in precalculus when dealing with vectors. Through the dot product, we can calculate the projection of one vector onto another, often revealing important geometric and physical properties. Mathematically, the dot product of two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \) in two-dimensional space is given by the formula:

• \( \mathbf{a} \cdot \mathbf{b} = (a_1 \cdot b_1) + (a_2 \cdot b_2) \)

This results in a scalar value, unlike vector addition or subtraction which yields another vector.
By using the dot product, you can determine if two vectors are perpendicular if their dot product is zero. This application is particularly useful in geometrical analysis and physics problems. To master dot products in precalculus, ensure you understand not just how to execute the calculation, but also why it is useful in analyzing the relationship between vectors.

Mathematics Behind Scalar Products

In mathematics, the scalar product offers insights beyond just computing values; it unveils relationships and angles between vectors. Because the dot product reduces two vector quantities to a single scalar, it transforms complex geometric scenarios into simpler one-dimensional problems. In our example,

• \( \mathbf{u} \cdot \mathbf{v} = -7 \)
• \( \mathbf{u} \cdot \mathbf{u} = 25 \)
• \( \mathbf{v} \cdot \mathbf{v} = 29 \)

The negative dot product signifies that vectors \( \mathbf{u} \) and \( \mathbf{v} \) are oriented in somewhat opposite directions. Meanwhile, \( \mathbf{u} \cdot \mathbf{u} = 25 \) and \( \mathbf{v} \cdot \mathbf{v} = 29 \) provide insights into the vectors' lengths or magnitudes, since these apply the dot product formula to the same vector.
Understanding these calculations offers a foundation for expansive mathematical topics such as vector spaces, transformations, and orthogonality. Therefore, harnessing the power of dot products extends your ability to critically approach and solve various mathematical puzzles with confidence.