Vector Basics: Understanding Directed Line Segments

A vector is a quantity that possesses both magnitude (length) and direction. It can be visually represented by a directed line segment, which has a distinct starting point known as the initial point and an ending point called the terminal point.

In this particular problem, we are provided with the coordinates of the initial point P and the terminal point Q in a three-dimensional (3D) space. Our primary objective is to determine the length of the vector formed by this directed line segment PQ.

Calculating Vector Components from Initial and Terminal Points

Before we can determine the length of the vector, we must first establish its components. If we have an initial point \(P(x_1, y_1, z_1)\) and a terminal point \(Q(x_2, y_2, z_2)\), the components of the vector \(\vec{PQ}\) are derived by subtracting the coordinates of the initial point from the corresponding coordinates of the terminal point.

The general formula for the components of a vector in a 3D coordinate system is:

\(\vec{PQ} = \langle x_2 - x_1, y_2 - y_1, z_2 - z_1 \rangle\)

Given the points:

• Initial point \(P(2, -3, 4)\)
• Terminal point \(Q(-2, 1, 1)\)

Now, let's calculate each component of the vector \(\vec{PQ}\):

• Component along the x-axis: \(x_2 - x_1 = -2 - 2 = -4\)
• Component along the y-axis: \(y_2 - y_1 = 1 - (-3) = 1 + 3 = 4\)
• Component along the z-axis: \(z_2 - z_1 = 1 - 4 = -3\)

Therefore, the vector \(\vec{PQ}\) can be expressed as \(\langle -4, 4, -3 \rangle\).

Length of the Vector: Determining the Magnitude

The length of the vector is also referred to as its magnitude. It represents the straight-line distance between the initial and terminal points of the directed line segment. For any vector \(\vec{v} = \langle a, b, c \rangle\) in a 3D space, its magnitude \(|\vec{v}|\) is computed using an extension of the Pythagorean theorem, often called the distance formula in 3D.

The formula for the magnitude of a vector in 3D space, using the initial and terminal points, is:

\(|\vec{PQ}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\)

Alternatively, if we already have the components \(\langle a, b, c \rangle\) of the vector, the magnitude is given by:

\(|\vec{PQ}| = \sqrt{a^2 + b^2 + c^2}\)

Using the components we previously calculated for \(\vec{PQ} = \langle -4, 4, -3 \rangle\):

\(|\vec{PQ}| = \sqrt{(-4)^2 + (4)^2 + (-3)^2}\)

First, calculate the squares of each component:

• \((-4)^2 = 16\)
• \((4)^2 = 16\)
• \((-3)^2 = 9\)

Now, sum these squared values:

\(|\vec{PQ}| = \sqrt{16 + 16 + 9}\)

\(|\vec{PQ}| = \sqrt{32 + 9}\)

\(|\vec{PQ}| = \sqrt{41}\)

Conclusion: The Length of the Directed Line Segment

The calculated length of the vector represented by the directed line segment with initial point P(2, -3, 4) and terminal point Q(-2, 1, 1) is \(\sqrt{41}\).