Problem 42 - Algebra and Trigonometry

Question

In Exercises \(39-46,\) find the unit vector that has the same direction as the vector \(\mathbf{v}\) $$\mathbf{v}=8 \mathbf{i}-6 \mathbf{j}$$

Step-by-Step Solution

Answer

Therefore, the unit vector that has the same direction as the vector \(\mathbf{v}=8 \mathbf{i}-6 \mathbf{j}\) is \(\mathbf{u}=0.8\mathbf{i}-0.6\mathbf{j}\)

Step 1: Calculating the magnitude of the vector

The magnitude of vector \(\mathbf{v}=8\mathbf{i}-6\mathbf{j}\) is calculated using the formula: \[\| \mathbf{v} \|=\sqrt{(8)^{2}+(-6)^{2}}\],which simplifies to \[\|\mathbf{v}\|=\sqrt{64+36}=\sqrt{100}=10\].

Step 2: Calculate the Unit Vector

The unit vector \(\mathbf{u}\) in the direction of \(\mathbf{v}\) is given by \[\mathbf{u}=\frac{\mathbf{v}}{\|\mathbf{v}\|}\].Substituting the values of \(\mathbf{v}\) and \(\|\mathbf{v}\|\) into the above equation, you get: \[\mathbf{u}=\frac{8\mathbf{i}-6\mathbf{j}}{10} = 0.8\mathbf{i} - 0.6\mathbf{j}\]

Key Concepts

Vector Magnitude

Understanding the magnitude of a vector is fundamental in physics and mathematics. In simple terms, the magnitude of a vector represents its length or size. To find it, one needs to consider a vector as a directed line segment in space, originating from a starting point and ending at a particular location.

For a two-dimensional vector like \(\mathbf{v}=8\mathbf{i}-6\mathbf{j}\), the magnitude, denoted by \(|| \mathbf{v} ||\), is found through the Pythagorean theorem. This involves squaring the components of the vector and taking the square root of their sum.

Following the formula \(|| \mathbf{v} ||=\sqrt{(8)^2+(-6)^2}\), the answer simplifies to \(||\mathbf{v}||=\sqrt{100}=10\), indicating that the vector's magnitude is 10. This step is crucial for calculating unit vectors, as it provides the denominator needed to normalize the original vector.

Vector Direction

While the magnitude of a vector tells us how 'long' the vector is, the direction indicates where the vector 'points'. It's an essential concept because vectors represent quantities that have both magnitude and direction, such as force or velocity.

To determine a vector's direction, we look at its components relative to the axes. For \(\mathbf{v}=8\mathbf{i}-6\mathbf{j}\), its direction is towards the positive x-axis and negative y-axis due to the positive 8 associated with \(\mathbf{i}\) (x-component) and the negative 6 associated with \(\mathbf{j}\) (y-component).

The direction can also be described using an angle measured from a specified axis, but in many applications, particularly in Cartesian coordinates, the focus is on the component form to understand in which quadrant the vector lies and its orientation with respect to standard axes.

Unit Vector

A unit vector is a vector that has a magnitude of '1'. It serves as the elemental vector that represents direction without any magnitude. The primary use of a unit vector is to express a vector in its normalized form, maintaining its direction but scaling its magnitude to 1.

To calculate the unit vector for \(\mathbf{v}=8\mathbf{i}-6\mathbf{j}\), first find the magnitude as shown previously. Then, you divide the original vector by its magnitude, which is \(||\mathbf{v}||=10\). Each component of the vector \(\mathbf{v}\) is divided by 10 to normalize it, yielding \(\mathbf{u}=\frac{8\mathbf{i}-6\mathbf{j}}{10} = 0.8\mathbf{i} - 0.6\mathbf{j}\).

This resultant unit vector \(\mathbf{u}\) points in the same direction as the original vector \(\mathbf{v}\) but with a reduced, uniform magnitude that makes it versatile for various mathematical and physical applications.