Question

Consider the following points \(P\) and \(Q\) a. Find \(\overrightarrow{P Q}\) and state your answer in two forms: \((a, b, c)\) and \(a i+b j+c \mathbf{k}\) b. Find the magnitude of \(\overrightarrow{P Q}\) c. Find two unit vectors parallel to \(\overrightarrow{P Q}\) $$P(1,5,0), Q(3,11,2)$$

Short answer

Question: Given points P(1, 5, 0) and Q(3, 11, 2), find the vector \(\overrightarrow{P Q}\) in triplet and \(i, j, k\) form, its magnitude, and two unit vectors parallel to it. Answer: The vector \(\overrightarrow{P Q}\) is (2, 6, 2) or \(2\mathbf{i} + 6\mathbf{j} + 2\mathbf{k}\). The magnitude of this vector is approximately 6.63. Two unit vectors parallel to \(\overrightarrow{P Q}\) are \(\frac{1}{\sqrt{44}}(2\mathbf{i}+6\mathbf{j}+2\mathbf{k})\) and \(-\frac{1}{\sqrt{44}}(2\mathbf{i}+6\mathbf{j}+2\mathbf{k})\).

Step-by-step solution

Find the vector \(\overrightarrow{P Q}\)

To find the vector \(\overrightarrow{P Q}\), subtract the coordinates of point \(P\) from the coordinates of point \(Q\). $$\overrightarrow{P Q} = Q - P = (3 - 1, 11 - 5, 2 - 0) = (2, 6, 2)$$

Answer in two forms

The vector can be represented as a triplet, and using \(i, j, k\) notation. Triplet: \((2, 6, 2)\) Using \(i, j, k\) notation: \(2\mathbf{i} + 6\mathbf{j} + 2\mathbf{k}\)

Find the magnitude of \(\overrightarrow{P Q}\)

We can find the magnitude of the vector \(\overrightarrow{P Q}\) using the formula: $$\left\lVert \overrightarrow{P Q} \right\rVert=\sqrt{a^2+b^2+c^2}$$ $$\left\lVert \overrightarrow{P Q} \right\rVert=\sqrt{2^2+6^2+2^2} = \sqrt{44} \approx 6.63$$

Find two unit vectors parallel to \(\overrightarrow{P Q}\)

To find two unit vectors parallel to \(\overrightarrow{P Q}\), we can divide the vector by its magnitude and multiply by the desired lengths. In this case, since we want unit vectors, we divide by the magnitude, and to find the second unit vector, we take the negative of the first one: $$Unit\,vector\,1 = \frac{1}{\sqrt{44}}(2\mathbf{i}+6\mathbf{j}+2\mathbf{k})$$ $$Unit\,vector\,2 = -\frac{1}{\sqrt{44}}(2\mathbf{i}+6\mathbf{j}+2\mathbf{k})$$

Key concepts

Vector Magnitude

Understanding the magnitude of a vector is crucial when working with vectors in 3D. The magnitude provides insight into the length of a vector regardless of its direction. If you imagine a vector as an arrow pointing from one point to another, the magnitude tells you how long this arrow is.
This is calculated using the formula:

• For a vector \( \vec{v} = (a, b, c) \), the magnitude is \( \left\lVert \vec{v} \right\rVert = \sqrt{a^2 + b^2 + c^2} \).

In the given exercise, the vector from point \( P \) to point \( Q \), denoted \( \overrightarrow{PQ} \), is \( (2, 6, 2) \).
We apply the magnitude formula here:

• \( \left\lVert \overrightarrow{PQ} \right\rVert = \sqrt{2^2 + 6^2 + 2^2} = \sqrt{44} \approx 6.63 \).

This calculation ensures you know exactly how long \( \overrightarrow{PQ} \) is in 3D space.

Unit Vectors

Unit vectors are vectors with a magnitude of one. They are important because they indicate direction alone without any emphasis on distance. When you need a vector that simply shows direction, unit vectors are what you use.
You can convert any vector into a unit vector by dividing it by its magnitude:

• If \( \vec{v} = (2, 6, 2) \) with a magnitude \( \sqrt{44} \), the unit vector \( \vec{u} \) is \( \frac{1}{\sqrt{44}}(2\mathbf{i} + 6\mathbf{j} + 2\mathbf{k}) \).

Sometimes, you also need a unit vector in the opposite direction. This is achieved simply by taking the negative of the unit vector:

• The second unit vector opposite to \( \vec{u} \) is \( -\frac{1}{\sqrt{44}}(2\mathbf{i} + 6\mathbf{j} + 2\mathbf{k}) \).

These unit vectors are strictly concerned with orientation and thus don't capture any notion of length beyond "one."

Vector Operations

Performing operations on vectors involves a few basic steps that allow you to manipulate vectors effectively. One common operation is finding the difference between two points in space to form a vector. For example, finding \( \overrightarrow{PQ} \) involves subtracting the coordinates of \( P \) from those of \( Q \), resulting in \( (2, 6, 2) \).
These operations can be represented in two forms:

• As a geometrical triplet: \( (2, 6, 2) \).
• Using unit vectors \( i, j, \text{and} \; k \): \( 2\mathbf{i} + 6\mathbf{j} + 2\mathbf{k} \).

Vector operations are not limited to subtraction. They can include addition, scaling, and more, providing a flexible toolkit for managing spatial data.
Determining which form to use can depend on context—geometrical representation can be intuitive for visualizing, while algebraic representation can simplify calculations.