Question

Find the unit vector in the same direction as \(\mathbf{V}\). \(\mathbf{v}=5 \mathbf{i}\)

Short answer

The unit vector is \(\textbf{i}\).

Step-by-step solution

Understand the Given Vector

The given vector is \(\textbf{v} = 5 \textbf{i}\). Since \(\textbf{i}\) is the unit vector in the x-direction, \(\textbf{v}\) is a vector in the same direction as the x-axis with a magnitude of 5.

Calculate the Magnitude of \(\textbf{v}\)

The magnitude of a vector \(\textbf{v} = a \textbf{i} + b \textbf{j} + c \textbf{k}\) is given by \(\textbar{}\textbf{v}\textbar{} = \sqrt{a^2 + b^2 + c^2}\). In this case, \(\textbf{v} = 5 \textbf{i}\), so the magnitude is \(\textbar{}\textbf{v}\textbar{} = \sqrt{5^2} = 5\).

Find the Unit Vector

To find the unit vector in the same direction as \(\textbf{v}\), divide \(\textbf{v}\) by its magnitude. Thus, the unit vector \(\textbf{u}\) is \(\textbf{u} = \frac{\textbf{v}}{\textbar{}\textbf{v}\textbar{}} = \frac{5 \textbf{i}}{5} = \textbf{i}\).

Key concepts

magnitude of a vector

When dealing with vectors, one of the key concepts is understanding their magnitude, also known as the vector's length. The magnitude of a vector tells us how long the vector is and is always a positive number. For a vector \(v\) given by \(v = a \textbf{i} + b j + c k\), its magnitude (\( \left| v \right| \)) can be calculated using the following formula:

\( \left| v \right| = \sqrt{a^2 + b^2 + c^2} \).

For the given problem's vector, which is \(v = 5 \textbf{i} \), the magnitude can be simplified as follows:

\( \left| v \right| = \sqrt{5^2} = 5 \).

This means the length of v is 5 units in the x-direction. Understanding the magnitude is crucial as it helps in subsequent calculations, such as finding the direction and the unit vector.

direction of a vector

Apart from the magnitude, a vector is also characterized by its direction. The direction of a vector tells us where the vector is pointing in the space. In a coordinate system, the direction can be determined using unit vectors along the axes- typically \(\textbf{i}\), \(\textbf{j}\), and \(\textbf{k}\), representing unit vectors in the x, y, and z directions, respectively.

In our given example, the vector \(v = 5 \textbf{i} \) points entirely in the x-direction. This simplifies the problem, as there is no component in the y or z directions. The direction of v is the same as that of \(\textbf{i}\).

Generally, to express the direction practically, we should find the unit vector, which simplifies understanding as it standardizes the length to 1 but keeps the direction intact. This leads us to our next concept: the unit vector calculation.

unit vector calculation

A unit vector is a vector that has a magnitude of exactly 1 unit and points in the same direction as the original vector. To find the unit vector in the direction of a given vector \(v\), we divide the vector by its magnitude.

Mathematically, if u is the unit vector in the direction of \(v\), it can be calculated as:

\(u = \frac{v}{\left| v \right|} \).

Applying this to our problem, where \(v = 5 \textbf{i}\) and \( \left| v \right| = 5 \), the unit vector u in the direction of \(v\) is given by:

\(u = \frac{5 \textbf{i}}{5}\).
This simplifies to:

\( u = \textbf{i}\).

Therefore, the unit vector in the same direction as \(v\) is simply \(d\)'s direction unit vector \(\textbf{i} \). Unit vectors are highly beneficial in various applications, including physics and engineering, for simplifying and standardizing direction-based calculations.