Suppose P = (0, – 1, 6) and PQ = (9, 2, – 4). and ( Enter an integer or decimal number (more.] Two more points on the line PQ are , ). (Enter the coordinates of two other points on the line PQ)
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
Suppose P=(0,-1,6) and −−→PQ(line)=⟨9,2,−4⟩.
Two more points on the line PQPQ are ((, , )) and (( , , , )).
(Enter the coordinates of two other points on the line PQPQ)
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### Sample Problem: Identifying Points on a Line in 3D Space
Suppose \( P = (0, -1, 6) \) and \(\overrightarrow{PQ} = (9, 2, -4) \).
**Task:**
Identify two more points on the line \( P Q \).
**Solution:**
Two more points on the line \( P Q \) are:
- \( ( \_ , \_ , \_ ) \)
- \( ( \_ , \_ , \_ ) \)
(Enter the coordinates of two other points on the line \( P Q \).)
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### Detailed Description
We are given a point \( P \) and a direction vector \(\overrightarrow{PQ}\). To find more points on the line, we can use the parametric form of the line equation. The parametric form of the line can be written as:
\[ \mathbf{r}(t) = \mathbf{P} + t \cdot \overrightarrow{PQ} \]
where \(\mathbf{P}\) is the starting point, \( t \) is a scalar parameter, and \(\overrightarrow{PQ}\) is the direction vector pointing from \( P \) to \( Q \).
For specific values of \( t \), calculate new points on the line:
1. **First Additional Point:**
Let \( t = 1 \),
\[
\mathbf{r}(1) = (0, -1, 6) + 1 \cdot (9, 2, -4) = (0 + 9, -1 + 2, 6 - 4) = (9, 1, 2)
\]
2. **Second Additional Point:**
Let \( t = 2 \),
\[
\mathbf{r}(2) = (0, -1, 6) + 2 \cdot (9, 2, -4) = (0 + 18, -1 + 4, 6 - 8) = (18, 3, -2)
\]
Therefore, two more points on the line \( P Q \) are:
- \( (9, 1, 2) \)
- \( (18, 3, -2) \)
(Ensure you replace the blanks with the above-calculated coordinates](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F85fb09c2-9b3d-4935-986b-649113c0a22e%2F3441bec0-c31c-49ef-be6e-25619f42d820%2Fh0sa7d.png&w=3840&q=75)
![**Title: Determining Coordinates on a Line in 3D Space**
**Instructions:**
Given two points, \( P \) and \( Q \), in a 3-dimensional space, we want to identify another point on the line passing through both \( P \) and \( Q \).
**Problem Statement:**
Suppose \( P = (-3, 5, -4) \) and \( Q = (5, 2, -10) \).
**Task:**
Calculate the coordinates of another point on the line \( PQ \).
**Answer Input:**
Another point on the line \( PQ \) is \( \left( \quad , \quad , \quad \right) \).
*(Enter the coordinates of another point on the line \( PQ \))*
**Explanation:**
- The coordinates \( P(x_1, y_1, z_1) = (-3, 5, -4) \) and \( Q(x_2, y_2, z_2) = (5, 2, -10) \) represent two distinct points on a line in 3D space.
- To find another point on this line, we can use parametric equations or simply extend the line segment using the directional vector from \( P \) to \( Q \). The vector from \( P \) to \( Q \) is given by:
\[
\overrightarrow{PQ} = Q - P = (5 - (-3), 2 - 5, -10 - (-4)) = (8, -3, -6)
\]
- By adding a scalar multiple of \(\overrightarrow{PQ}\) to \(P\), we can find another point on the line. For example, for the parameter \( t = 1 \):
\[
R = P + t \cdot \overrightarrow{PQ} = (-3, 5, -4) + 1 \cdot (8, -3, -6) = (-3 + 8, 5 - 3, -4 - 6) = (5, 2, -10)
\]
Confirming \(Q\). To get a different point, you can try \( t = \frac{1}{2} \):
\[
R = P + \frac{1}{2} \cdot \overrightarrow{PQ} =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F85fb09c2-9b3d-4935-986b-649113c0a22e%2F3441bec0-c31c-49ef-be6e-25619f42d820%2Fkv2wiu.png&w=3840&q=75)
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