Problem 53 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The unit vector is \(\frac{1}{\backslashsqrt{2}} \backslashmathbf{i} - \frac{1}{\backslashsqrt{2}} \backslashmathbf{j}\)

Step 1: Find the Magnitude of the Vector

Calculate the magnitude of vector \(\backslashmathbf{v} = \backslashmathbf{i} - \backslashmathbf{j}\). The magnitude is given by the formula \[\|\backslashmathbf{v}\| = \backslashsqrt{v_1^2 + v_2^2}\]. In this case, \backslashmathbf{v} = 1\backslashi - 1\backslashj\, so we need to compute: \[\backslash|\backslashmathbf{v}\backslash| = \backslashsqrt{1^2 + (-1)^2} = \backslashsqrt{2}\].

Step 2: Use the Magnitude to Find the Unit Vector

A unit vector \(\backslashmathbf{u}\) in the direction of \(\backslashmathbf{v}\) is found by dividing each component of \(\backslashmathbf{v}\) by its magnitude. Therefore, \(\backslashmathbf{u} = \frac{1}{\backslashsqrt{2}} \backslashmathbf{i} - \frac{1}{\backslashsqrt{2}} \backslashmathbf{j}\)

Key Concepts

Vector Magnitude

The magnitude of a vector is like the length of the vector. To find the magnitude, use the Pythagorean theorem. For a 2-dimensional vector \(\backslashmathbf{v} = \backslashmathbf{i} - \backslashmathbf{j}\), we calculate it as \[\|\backslashmathbf{v}\| = \sqrt{v_1^2 + v_2^2}\].So, for \(\backslashmathbf{v} = 1\backslashi - 1\backslashj\), it becomes \[\|\backslashmathbf{v}\| = \sqrt{1^2 + (-1)^2} = \sqrt{2}\]. This is because both individual components of the vector get squared and then summed before taking the square root. It's important to understand the magnitude because it tells how 'big' the vector is.

Vector Components

Vectors are defined by their components along various axes. In two-dimensional space, these are usually the x (i) and y (j) components. For \(\backslashmathbf{v} = 1\backslashi - 1\backslashj\):

• The x-component is \(1\backslashi\).
• The y-component is \(-1\backslashj\).

These components show how far and in which direction the vector moves along each axis. Knowing the components helps in calculating the vector's magnitude as well as the unit vector.

Unit Vector Calculation

A unit vector has a magnitude of 1 and points in the same direction as the original vector. To find the unit vector, divide each component of the vector by its magnitude. For \(\backslashmathbf{v} = 1\backslashi - 1\backslashj\) with magnitude \(\sqrt{2}\), the unit vector \(\backslashmathbf{u}\) is:

• For x-component: \(\frac{1}{\sqrt{2}}\).
• For y-component: \(\frac{-1}{\sqrt{2}}\).

This simplifies to: \[\backslashmathbf{u} = \frac{1}{\sqrt{2}} \backslashi - \frac{1}{\sqrt{2}} \backslashj\]. Unit vectors are helpful for indicating direction without regard to length. They are also fundamental in vector normalization processes.