Problem 9 - Contemporary Precalculus

Question

Find the dot product when \(u=\langle 4,3\rangle\) \(\boldsymbol{v}=\langle-5,2\rangle,\) and \(\boldsymbol{w}=\langle 4,-1\rangle\) $$(\mathbf{u}+\mathbf{v}) \cdot(\mathbf{v}+\mathbf{w})$$

Step-by-Step Solution

Answer

Answer: The dot product of (u + v) and (v + w) is 6.

Step 1: Add vectors u and v

To find (u + v), we need to add their corresponding components. For u = <4, 3> and v = <-5, 2>, we have: (u + v) = <4 + (-5), 3 + 2> = <-1, 5>.

Step 2: Add vectors v and w

To find (v + w), we need to add their corresponding components. For v = <-5, 2> and w = <4, -1>, we have: (v + w) = <-5 + 4, 2 + (-1)> = <-1, 1>.

Step 3: Calculate the dot product

Now, we need to find the dot product of (u + v) and (v + w). The dot product is calculated as follows: (u + v) ● (v + w) = (-1) * (-1) + 5 * 1 = 1 + 5 = 6. So, the dot product when (u + v) and (v + w) is 6.

Key Concepts

Vector Addition

Vector addition is a fundamental operation in precalculus mathematics involving vectors, which are quantities that have both magnitude and direction. When we add vectors, we operate component-wise, which means we add the corresponding components (horizontal and vertical parts) of each vector together.

Consider two vectors, \( \textbf{a} = \langle a_1, a_2 \rangle \) and \( \textbf{b} = \langle b_1, b_2 \rangle \). The sum \( \textbf{a} + \textbf{b} \) is given by adding the corresponding components of these vectors: \( \textbf{a} + \textbf{b} = \langle a_1 + b_1, a_2 + b_2 \rangle \).

For example, if \( \textbf{u} = \langle 4, 3 \rangle \) and \( \textbf{v} = \langle -5, 2 \rangle \), their sum is \( \textbf{u} + \textbf{v} = \langle 4 + (-5), 3 + 2 \rangle = \langle -1, 5 \rangle \).

• Component-wise Addition: The essence of vector addition is to add the vectors' components separately.
• Graphical Interpretation: Visually, vector addition can be represented by joining the tail of one vector to the head of another, often using the 'tip-to-tail' method.
• Applications: This operation is not just an abstract concept; it has practical implications in physics, engineering, and computer graphics.

Vector addition is commutative, meaning that the order in which you add vectors does not affect the result.

Vector Operations

Vector operations encompass a range of mathematical procedures applied to vectors. These operations include addition, subtraction, scalar multiplication, and the dot product, to name a few.

• Addition and Subtraction: As we covered in vector addition, these operations are performed component-wise, meaning corresponding components are added or subtracted.
• Scalar Multiplication: Multiplying a vector by a scalar (a real number) involves multiplying each component of the vector by the scalar, effectively scaling the vector's magnitude.
• Dot Product: A critical operation wherein the dot product of vectors \( \textbf{a} \) and \( \textbf{b} \) is computed as \( \textbf{a} \cdot \textbf{b} = a_1b_1 + a_2b_2 \) for two-dimensional vectors. This operation results in a scalar.

The dot product, also known as the scalar product, has significant geometrical interpretation: it can be used to determine the angle between vectors and to check if vectors are perpendicular. In the context of the given exercise, finding the dot product of \( (\textbf{u} + \textbf{v}) \cdot (\textbf{v} + \textbf{w}) \) requires adding the vectors first and then using their components to find the product: \(-1\cdot-1 + 5\cdot1 = 1 + 5 = 6\).

Practical Importance of Vector Operations

These operations are not just for theoretical purposes; they are essential in fields such as physics for representing forces, velocities, and much more, and in computer graphics for manipulating graphics and animations.

Precalculus Mathematics

Precalculus mathematics is a course that lays the foundation for calculus and other higher-level mathematics courses. It encompasses a wide range of topics including functions, polynomials, exponents, logarithms, and of course, vectors.

Vectors are especially important as they bridge algebra and geometry, generating a deep understanding of multi-dimensional concepts. The concepts of vector addition and operations, as shown in this exercise, are critical precalculus elements.

The dot product, for example, is a key precalculus concept because not only does it require an understanding of vector operations, but also sets the stage for further study in vector calculus and analysis. Moreover, mastering precalculus ensures a smoother transition into calculus since it introduces limits and continuity concepts through the detailed study of functions.

How Precalculus Applies to Real-World Problems

Understanding vectors and their operations is not merely academic; these principles are applied in physics to describe forces and motions, in engineering for designing structures, and in economics for optimizing functions. The computation done in the exercise represents a basic yet powerful application of precalculus concepts to solve real-world problems.