Problem 1

Question

Find the parametric equations of the line through the given pair of points. $(1,-2,3),(4,5,6)$

Step-by-Step Solution

Verified

Answer

Parametric equations: $x = 1 + 3t$, $y = -2 + 7t$, $z = 3 + 3t$.

1Step 1: Identify Direction Vector

To find the parametric equations of a line, we first need the direction vector. Given points are $(1, -2, 3)$ and $(4, 5, 6)$. The direction vector $\mathbf{v}$ is calculated by subtracting the coordinates of the first point from the second point: \[ \mathbf{v} = (4 - 1, 5 - (-2), 6 - 3) = (3, 7, 3) \]

2Step 2: Write Parametric Equations

Next, use the point $(1, -2, 3)$ as a point on the line, and the vector $(3, 7, 3)$ as the direction vector. The parametric equations are expressed as: \[ x = 1 + 3t \] \[ y = -2 + 7t \] \[ z = 3 + 3t \] Here, $t$ is the parameter and can be any real number.

Key Concepts

Direction VectorPoints in 3D SpaceLine through PointsVector Subtraction

Direction Vector

A direction vector is essential when defining a line in three-dimensional space using parametric equations. It tells us the direction in which the line extends. To find the direction vector, we subtract the coordinates of one given point from the other. For example, given the points

$(1, -2, 3)$
$(4, 5, 6)$\

, we calculate the direction vector as follows: \[ (4-1, 5-(-2), 6-3) \]. The result is the vector $(3, 7, 3)$.
This vector signifies that for every movement from one point to another along the line, the x-coordinate increases by 3, the y-coordinate by 7, and the z-coordinate by 3.
Such calculations are crucial for understanding the geometry of the line in 3D coordinates.

Points in 3D Space

Points in 3D space are identified by coordinates, represented as $(x, y, z)$, where $x$, $y$, and $z$ denote positions along the respective axis.

Coordinate $x$ determines the horizontal position.
Coordinate $y$ indicates the vertical position.
Coordinate $z$ dictates the depth at that position.

These components together define any point uniquely in three-dimensional space. Understanding how to manipulate and interpret these coordinates is foundational for solving problems in vector calculus and applying parametric equations for lines and other spatial relationships. For example, given points such as

$(1, -2, 3)$
$(4, 5, 6)$\

helps us to establish the spatial relationship and direction of lines passing through them.

Line through Points

A line in 3D space can be described in parametric form through given points. Such a description requires an initial point known as the position point and a direction vector.
A parametric equation represents each component (x, y, z) in terms of a parameter, usually denoted by $t$. This equation is helpful when you need to express a line passing through a specific set of points. Using the points

$(1, -2, 3)$
$(4, 5, 6)$\

and the direction vector

$(3, 7, 3)$

, the parametric equations of the line are:

x-coordinate: $x = 1 + 3t$
y-coordinate: $y = -2 + 7t$
z-coordinate: $z = 3 + 3t$

These equations detail how each coordinate of the point on the line varies with $t$. Here, $t$ is a real number, representing different points along the line.

Vector Subtraction

Vector subtraction is a mathematical operation used to determine the vector between two points in space.
This operation is integral when finding the direction vector of a line. When subtracting two vectors, each of their corresponding components are subtracted.
For the points