Question

Write an expression in rectangular components for the vector that extends from \(\left(x_{1}, y_{1}, z_{1}\right)\) to \(\left(x_{2}, y_{2}, z_{2}\right)\) and determine the magnitude of this vector.

Short answer

Question: Given two points in 3D space, (3, -1, 4) and (6, 2, 1), find the vector that extends from the first point to the second point and determine its magnitude. Answer: To find the vector extending from (3, -1, 4) to (6, 2, 1), follow the steps outlined in the solution. Calculate the differences in rectangular components: Δx = 6 - 3 = 3 Δy = 2 - (-1) = 3 Δz = 1 - 4 = -3 Combine the differences into a single vector: vec(v) = ⟨3, 3, -3⟩ Determine the magnitude of the vector: |vec(v)| = √(3^2 + 3^2 + (-3)^2) = √27 Therefore, the vector extending from (3, -1, 4) to (6, 2, 1) is vec(v) = ⟨3, 3, -3⟩, and its magnitude is |vec(v)| = √27.

Step-by-step solution

Write down the problem

Given two points in 3D space, \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\), we need to find the vector that extends from the first point to the second point and then find its magnitude.

Find the difference in rectangular components

To find the vector extending from \((x_1, y_1, z_1)\) to \((x_2, y_2, z_2)\), we need to find the difference of their respective rectangular components. This can be done as follows: $$ \begin{aligned} \Delta x &= x_2 - x_1 \\ \Delta y &= y_2 - y_1 \\ \Delta z &= z_2 - z_1 \end{aligned} $$

Combine the differences into a single vector

Now that we have the difference in rectangular components, we can represent the vector extending from \((x_1, y_1, z_1)\) to \((x_2, y_2, z_2)\) as follows: $$ \vec{v} = \left\langle \Delta x, \Delta y, \Delta z \right\rangle $$

Determine the magnitude of the vector

Now that we have the vector \(\vec{v}\) in terms of its rectangular components, we can determine its magnitude using the Euclidean distance formula for a 3D space: $$ \lvert\vec{v}\rvert = \sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2} $$ Following these steps, we will be able to find the expression of the vector in rectangular components and determine its magnitude.

Key concepts

Rectangular Components

When dealing with vectors in 3D space, it's crucial to understand rectangular components. A vector extending from one point \( (x_1, y_1, z_1) \) to another point \( (x_2, y_2, z_2) \) is all about connecting these coordinates. Here's how that works:
To find each part of the vector, subtract the coordinates of the starting point from the coordinates of the ending point:

• For the x-component: \( x_2 - x_1 \).
• For the y-component: \( y_2 - y_1 \).
• For the z-component: \( z_2 - z_1 \).

These differences are called the rectangles or rectangular components of the vector. By combining them, we form the vector \( \vec{v} = \langle \Delta x, \Delta y, \Delta z \rangle \). This sets the stage for defining the vector's direction and magnitude.

Vector Magnitude

Vector magnitude is like measuring the length of a vector. It helps us understand how long the vector is in space, no matter where it points. Once we have a vector expressed in rectangular components \( \vec{v} = \langle \Delta x, \Delta y, \Delta z \rangle \), we can find its magnitude or length.
The magnitude tells us how far apart the two points are in 3D space. The formula for finding the magnitude of a vector \( \vec{v} \) is:\[ \lvert \vec{v} \rvert = \sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2} \]This formula might look similar to the Pythagorean theorem. That's because it extends this idea into three dimensions, taking into account all three components of the vector. The result is always a non-negative number that represents the vector's length.

Euclidean Distance Formula

The Euclidean distance formula is a powerful tool for measuring the "straight-line" distance between two points in space. It's especially handy when these points are in a 3-dimensional space.
Here's how it relates to vectors:

• Using starting point \( (x_1, y_1, z_1) \) and ending point \( (x_2, y_2, z_2) \)
• The formula becomes the magnitude of the vector connecting these points.
• This gives: \[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

This formula allows us to calculate how far apart these two points are, giving a clear numerical value to their separation in space. Whether used in physics, engineering, or computer graphics, it’s a fundamental principle that helps us navigate and measure the 3D world.