Question

Find vectors parallel to \(\mathbf{v}\) of the given length. $$\mathbf{v}=\overrightarrow{P Q} \text { with } P(3,4,0) \text { and } Q(2,3,1) ; \text { length }=3$$

Short answer

Question: Find a vector parallel to the vector passing through points P(3, 4, 0) and Q(2, 3, 1) with a length of 3. Solution: The vector parallel to the given vector with a length of 3 is \(\mathbf{w} = \begin{pmatrix} -\sqrt{3} \\ -\sqrt{3} \\ \sqrt{3} \end{pmatrix}\).

Step-by-step solution

1. Find the coordinates of vector \(\mathbf{v}\)

To find the coordinates of vector \(\mathbf{v}\), we subtract the coordinates of point P from the coordinates of point Q: $$ \mathbf{v} = \begin{pmatrix} Q_x - P_x \\ Q_y - P_y \\ Q_z - P_z \end{pmatrix}=\begin{pmatrix} 2-3 \\ 3-4 \\ 1-0 \end{pmatrix} = \begin{pmatrix} -1 \\ -1 \\ 1 \end{pmatrix} $$

2. Calculate the length of vector \(\mathbf{v}\)

To calculate the length of vector \(\mathbf{v}\), we use the Euclidean norm formula: $$ \|\mathbf{v}\|=\sqrt{(-1)^2+(-1)^2+1^2}=\sqrt{1+1+1}=\sqrt{3} $$

3. Find the unit vector parallel to \(\mathbf{v}\)

To find the unit vector, we need to divide each coordinate of vector \(\mathbf{v}\) by its length: $$ \mathbf{u}=\frac{1}{\|\mathbf{v}\|}\mathbf{v}=\frac{1}{\sqrt{3}}\begin{pmatrix}-1 \\ -1 \\ 1\end{pmatrix}=\begin{pmatrix}-\frac{1}{\sqrt{3}} \\ -\frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}}\end{pmatrix} $$

4. Scale the unit vector to the desired length

Now, we will multiply each coordinate of the unit vector by the desired length (3): $$ \mathbf{w}=3\mathbf{u}=3\begin{pmatrix}-\frac{1}{\sqrt{3}} \\ -\frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}}\end{pmatrix}=\begin{pmatrix}-\sqrt{3} \\ -\sqrt{3} \\ \sqrt{3}\end{pmatrix} $$ The vector \(\mathbf{w}\) is parallel to the original vector \(\mathbf{v}\) and has a length of 3.

Key concepts

Finding Parallel Vectors

When trying to find a vector parallel to a given vector, the essential property that needs to be maintained is direction. Two vectors are parallel if one is a scalar multiple of the other. In the provided exercise, the goal is to find a vector that is parallel to vector \(\textbf{v}\) of a given length. To start with, we determine the directional vector by subtracting the coordinates of point P from point Q, which results in vector \(\textbf{v}\). However, to ensure the new vector is both parallel and has the required length, we must first normalize \(\textbf{v}\) to create a unit vector, then scale it to the desired magnitude, as shown in steps 3 and 4 of the solution.

This process is indispensable in physics and engineering when we need to maintain direction but adjust magnitude, such as when scaling forces or velocities.

Vector Length (Magnitude)

The length or magnitude of a vector provides a quantitative measure of how long it is. In the context of the exercise, we calculated the magnitude of vector \(\textbf{v}\) using the Euclidean norm, which involves taking the square root of the sum of the squares of the vector's components. The formula for the magnitude of a vector \(\textbf{v} = (v_x, v_y, v_z)\) in three-dimensional space is:

\[\|\textbf{v}\| = \sqrt{v_x^2 + v_y^2 + v_z^2}\]

Understanding how to compute vector magnitudes is fundamental in calculus, as it is closely related to the concept of distance in space, and is frequently utilized when dealing with vector fields, force vectors, and even in computer graphics algorithms for normalization.

Unit Vector Calculation

A unit vector is a vector that has a magnitude of one, maintaining the direction of the original vector but discarding its magnitude. To find the unit vector parallel to any given vector, we divide each component of the original vector by its magnitude. The unit vector of vector \(\textbf{v}\) is denoted as \(\textbf{u}\), and it is calculated as:\[\textbf{u} = \frac{1}{\|\textbf{v}\|}\textbf{v}\]

This creates a dimensionless vector that serves as a basis for creating other vectors of desired lengths in the same direction as the original. Being able to calculate unit vectors is a cornerstone in working with vectors as it simplifies the process of altering vector magnitudes and is especially useful in 3D modeling and physics to denote directional properties without scale.

Euclidean Norm

The Euclidean norm is a measure of vector magnitude rooted in Euclidean geometry and is also known as the L2 norm. It's applicable in multiple dimensions and is defined as the square root of the sum of the squares of a vector's components. Mathematically, given a vector \(\textbf{v}=(v_1, v_2, \ldots, v_n)\) in \(n\)-dimensional space, the Euclidean norm is given by:\[\|\textbf{v}\| = \sqrt{v_1^2 + v_2^2 + \ldots + v_n^2}\]

It's akin to the Pythagorean theorem extended into n-dimensions and is integral to distance metrics in spaces of any dimensionality. The Euclidean norm provides the base for various operations in calculus, optimization, and computational geometry.