Problem 4 - Active Calculus Multivariable

Question

Resolve the following vectors into components: (a) The vector \(\vec{v}\) in 2 -space of length 5 pointing up at an angle of \(\pi / 4\) measured from the positive \(x\) -axis. \vec{v}=________\(\vec{i}+\)________\(\vec{j}\) (b) The vector \(\vec{w}\) in 3 -space of length 3 lying in the \(x z\) -plane pointing upward at an angle of \(2 \pi / 3\) measured from the positive \(x\) -axis. \vec{v}=________\(\vec{i}+\)________\(\vec{j}\)+________\(\vec{k}\)

Step-by-Step Solution

Answer

\(\vec{v} = 5 \cos (\pi / 4)\hat{i} + 5 \sin (\pi / 4)\hat{j}\) \(\vec{w} = 3 \cos (2\pi / 3)\hat{i} + 0\hat{j} + 3 \sin (2\pi / 3)\hat{k}\)

Step 1: Find the x-component of \(\vec{v}\)

To find the x-component of \(\vec{v}\), we can use the cosine function. The x-component of \(\vec{v}\) is equal to the length of \(\vec{v}\) times the cosine of the angle measured from the positive x-axis. This gives us: \[v_{x} = 5 \cos (\pi / 4)\]

Step 2: Find the y-component of \(\vec{v}\)

To find the y-component, we can use the sine function. The y-component of \(\vec{v}\) is equal to the length of \(\vec{v}\) times the sine of the angle measured from the positive x-axis. This gives us: \[v_{y} = 5 \sin (\pi / 4)\]

Step 3: Write \(\vec{v}\) as a linear combination of unit vectors

We now have the x and y components of \(\vec{v}\). To express it as a linear combination of unit vectors, we have: \[\vec{v} = 5 \cos (\pi / 4)\hat{i} + 5 \sin (\pi / 4)\hat{j}\] (b) Resolve the vector \(\vec{w}\) in 3-space:

Step 4: Find the x-component of \(\vec{w}\)

To find the x-component of \(\vec{w}\), we can use the cosine function. The x-component of \(\vec{w}\) is equal to the length of \(\vec{w}\) times the cosine of the angle measured from the positive x-axis. This gives us: \[w_{x} = 3 \cos (2\pi / 3)\]

Step 5: Find the z-component of \(\vec{w}\)

To find the z-component of \(\vec{w}\), we can use the sine function. The z-component of \(\vec{w}\) is equal to the length of \(\vec{w}\) times the sine of the angle measured from the positive x-axis. This gives us: \[w_{z} = 3 \sin (2\pi / 3)\]

Step 6: Write \(\vec{w}\) as a linear combination of unit vectors

We now have the x and z components of \(\vec{w}\). Since \(\vec{w}\) lies in the xz-plane, its y-component will be zero. To express it as a linear combination of unit vectors, we have: \[\vec{w} = 3 \cos (2\pi / 3)\hat{i} + 0\hat{j} + 3 \sin (2\pi / 3)\hat{k}\]

Key Concepts

Vector Resolution

Vector resolution is a crucial concept in physics and engineering that allows us to break down a vector into its constituent components along specified axes. Imagine you're playing tug-of-war, and the rope is angled; vector resolution would be like finding out how much of your pulling power is going directly forward and how much is going upwards.

In the given exercise, we resolve vector \(\vec{v}\) into x and y components in a 2-dimensional space, and vector \(\vec{w}\) into x, y, and z components in a 3-dimensional space. The process involves using trigonometric functions to determine the magnitude of each component based on the angle the vector makes with a reference axis. Using trigonometry simplifies the process, allowing precise calculations of each component's value.

Trigonometric Functions

Trigonometric functions are the bridge between angles and side lengths in triangles, and they are a foundation for resolving vectors into components. The most common functions we use are sine (sin) and cosine (cos).

In our exercise, the cosine function determines the x-component of a vector based on its angle from the positive x-axis, while the sine function does so for the y-component. For example, a vector pointing at an angle \(\pi / 4\) would have equal x and y components because sine and cosine of \(\pi / 4\) are equal, which reflects the vector's equal projection onto both axes.

Linear Combination of Unit Vectors

A linear combination of unit vectors expresses how we add scaled versions of these vectors together to create another vector. The unit vectors \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\) represent one unit of length in the x, y, and z directions, respectively.

To construct the vector from its components like we did in the exercise, we multiply each unit vector by the corresponding component magnitude. This operation yields a representation of the original vector in a coordinate system, providing a clear geometric interpretation.

Multivariable Calculus

Multivariable calculus extends the concepts of calculus to functions with more than one variable. This branch of mathematics is essential when dealing with vectors in space as it allows us to navigate and express phenomena in two or three dimensions, providing the tools to understand curves, surfaces, and even changes in these entities.

When we work with vector components and resolve them in space, we touch the surface of multivariable calculus. Vector functions, partial derivatives, and gradients are all part of multivariable calculus and play a significant role in deeper analyses of vector behavior in space.