Problem 87 - Precalculus

Question

There is a branch of calculus devoted to the study of vectorvalued functions; these are functions that map real numbers onto vectors. For example, \(v(t)=\langle t, 2 t\rangle\). Calculate the dot product of the vector-valued functions \(\mathbf{u}(t)=\left\langle 2 t, t^{2}\right\rangle\) and \(\mathbf{v}(t)=\langle t,-3 t\rangle\).

Step-by-Step Solution

Answer

The dot product of \( \mathbf{u}(t) \) and \( \mathbf{v}(t) \) is \( 2t^2 - 3t^3 \).

Step 1: Define the Dot Product Formula

The dot product of two vectors \( \mathbf{u}(t) = \langle a_1, a_2 \rangle \) and \( \mathbf{v}(t) = \langle b_1, b_2 \rangle \) is calculated as \( a_1b_1 + a_2b_2 \). To find the dot product of vector-valued functions \( \mathbf{u}(t) \) and \( \mathbf{v}(t) \), apply this formula.

Step 2: Identify Components of the Vectors

For \( \mathbf{u}(t) = \langle 2t, t^2 \rangle \), the components are \( a_1 = 2t \) and \( a_2 = t^2 \). For \( \mathbf{v}(t) = \langle t, -3t \rangle \), the components are \( b_1 = t \) and \( b_2 = -3t \).

Step 3: Calculate the Product of Corresponding Components

First, calculate \( a_1b_1 = (2t) \cdot t = 2t^2 \). Then, calculate \( a_2b_2 = (t^2) \cdot (-3t) = -3t^3 \).

Step 4: Sum the Products

Add the results from Step 3 to find the dot product: \( 2t^2 + (-3t^3) = 2t^2 - 3t^3 \).

Step 5: Write the Final Answer

The dot product of the vector-valued functions \( \mathbf{u}(t) \) and \( \mathbf{v}(t) \) is \( 2t^2 - 3t^3 \).

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a crucial operation in vector calculus, especially when dealing with vectors. Normally, it helps us measure the angle and relation between two vectors. To find the dot product, we need two vectors, say \( \mathbf{u} = \langle a_1, a_2 \rangle \) and \( \mathbf{v} = \langle b_1, b_2 \rangle \). The dot product sums the products of corresponding components: \( a_1 \cdot b_1 + a_2 \cdot b_2 \). This process turns a pair of vectors into a single real number.

Why is the dot product important? It provides a way to calculate the projection of one vector onto another, which is widely used in physics for concepts like work and in computer graphics for shading and lighting models. The dot product is zero if vectors are orthogonal, indicating they are at right angles to each other.

Vector-Valued Functions

Vector-valued functions are functions that associate real numbers with vectors rather than scalars. They map an interval of real numbers onto a series of vectors and can describe paths in a plane or space.

For instance, if we consider \( \mathbf{u}(t) = \langle 2t, t^2 \rangle \), as \( t \) varies, this function generates a series of vectors that represent a path in a 2-dimensional plane. Such functions are significant in motion and physics problems where you need to track the position of a moving object over time.

By using vector-valued functions, you can compactly express the movement of an object through its components, making it easier to analyze quantities like velocity and acceleration by taking derivatives.

Real Numbers to Vectors

Converting real numbers into vectors is a central concept in vector calculus, particularly when dealing with vector-valued functions. Each real number from the domain can be seen as a parameter controlling the vector's components. For example, \( v(t)=\langle t, 2t \rangle \) takes a real number \( t \) and outputs a vector \( \langle t, 2t \rangle \).

This mapping from real numbers to vectors helps describe more complex geometries and motions than what is possible with real functions alone. It allows us to observe how the outputs (vectors) of functions change as the inputs (real numbers) vary.

From physics to computer graphics, the ability to translate simple numbers into dynamic, multi-dimensional data points is invaluable for simulating and understanding real-world phenomena.