Problem 48 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1.$$\mathbf{u}=\langle 0,-2\rangle$$.

Step-by-Step Solution

Answer

The unit vector in the direction of the vector \( \mathbf{u}=\langle 0,-2\rangle \) is \( \langle 0,-1 \rangle \).

Step 1: Find the magnitude of the vector

First, we need to find the magnitude of the vector \( \mathbf{u}=\langle 0,-2\rangle \) using the formula \( \sqrt{a^2 + b^2} \). The magnitude of \( \mathbf{u} \) is \( \sqrt{0^2 + (-2)^2} = 2 \).

Step 2: Find the unit vector

Next, divide vector \( \mathbf{u} \) by its magnitude to obtain the unit vector. This gives us \( \langle \frac{0}{2}, \frac{-2}{2} \rangle = \langle 0,-1 \rangle \).

Step 3: Verify the magnitude of the unit vector

Lastly, we need to verify that the magnitude of the unit vector \( \langle 0,-1 \rangle \) is 1. This translates to \( \sqrt{0^2 + (-1)^2} = 1 \), confirming our calculation.

Key Concepts

Vector Magnitude

The magnitude of a vector is essentially its length. To find the magnitude, we apply the Pythagorean Theorem to the components of the vector. For any two-dimensional vector \( \mathbf{v} = \langle a, b \rangle \), the magnitude \( |\mathbf{v}| \) is calculated using the formula \( \sqrt{a^2 + b^2} \). This formula helps us determine how "long" the vector is in a geometrical sense.

In our example with the vector \( \mathbf{u} = \langle 0, -2 \rangle \):

• The value for \( a \) is 0.
• The value for \( b \) is -2.

Substituting these values into the formula gives us \( \sqrt{0^2 + (-2)^2} = \sqrt{4} = 2 \). So, the magnitude of the vector \( \mathbf{u} \) is 2.

Understanding the magnitude is crucial when dealing with vectors, as it provides insights into the size of the vector, which is important for further calculations, like finding the unit vector.

Unit Vector Calculation

Once the magnitude of a vector is known, calculating the unit vector becomes straightforward. A unit vector has a magnitude of 1 and points in the same direction as the original vector. It is essentially a convenient way to express direction without considering length.

To find the unit vector of \( \mathbf{u} = \langle 0, -2 \rangle \), we divide each component of the vector by its magnitude. Here’s the step-by-step process:

• The magnitude of our vector \( \mathbf{u} \) is 2, as calculated previously.
• Divide each component of the vector by this magnitude:
  \( \langle \frac{0}{2}, \frac{-2}{2} \rangle = \langle 0, -1 \rangle \).

This result, \( \langle 0, -1 \rangle \), is the unit vector. It has a magnitude of 1, and it points in the same direction as the vector \( \mathbf{u} \).

Unit vectors are particularly useful in physics and engineering, where they help in describing directions of forces or fields while focusing on their direction rather than their size.

Vector Direction

In vector algebra, direction plays a critical role in understanding how a vector behaves in space. The direction of a vector is specified by its components, indicating how far the vector extends along each axis from the origin point.

For the vector \( \mathbf{u} = \langle 0, -2 \rangle \), the components are:

• 0 units in the x-direction (horizontal).
• -2 units in the y-direction (vertical).

This means the vector points directly downwards from the origin, as it does not move sideways (x-direction is zero).

Unit vectors help in defining direction more precisely and clearly, quality especially needed in vector operations, transformations, and applications in various fields like computer graphics and physics. By translating a vector to its unit form, you express the vector's direction in a standardized way, excluding the influence size or scale has, thus making calculations easier and results more interpretable.