Problem 56 - 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. $$\mathbf{w}=-3 \mathbf{i}$$.

Step-by-Step Solution

Answer

The unit vector in the direction of \( \mathbf{w} = -3 \mathbf{i} \) is \( - \mathbf{i} \).

Step 1: Understanding the unit vector

A unit vector is a vector of length 1. Another important fact is that a unit vector in the direction of a given vector can be obtained by dividing that vector by its magnitude.

Step 2: Finding the magnitude of the given vector

Given the vector \( \mathbf{w} = -3 \mathbf{i} \), the magnitude \(|\mathbf{w}|\) of this vector is the absolute value of its coefficient since it's a one-dimensional vector, so \(|\mathbf{w}|= |-3| = 3\).

Step 3: Finding the unit vector

The unit vector in the direction of \( \mathbf{w} \) is obtained by dividing \( \mathbf{w} \) by its magnitude. So, \( \mathbf{u}_w = \frac{\mathbf{w}}{|\mathbf{w}|} = \frac{-3 \mathbf{i}}{3} = - \mathbf{i}\).

Step 4: Verifying the magnitude

Let’s verify the magnitude of calculated unit vector is 1. The magnitude of the vector \( - \mathbf{i} \) is |-1| = 1, which is in accordance with the definition of a unit vector.

Key Concepts

Vector Magnitude

The magnitude of a vector represents its length or size. It is a crucial concept when dealing with vectors as it helps determine how long or large the vector is in its respective space. When calculating the magnitude, you consider the vector as a geometric object that has both direction and magnitude. For a one-dimensional vector like \( \mathbf{w} = -3 \mathbf{i} \), determining the magnitude is straightforward because it involves only one component. For such vectors, the magnitude \(|\mathbf{w}|\) is simply the absolute value of the vector's coefficient. Hence, \(|\mathbf{w}| = |-3| = 3\). For more complex vectors with multiple dimensions, the magnitude is calculated using the Pythagorean theorem by taking the square root of the sum of the squares of each component.

Direction of Vector

Understanding the direction of a vector is as important as knowing its magnitude. The direction indicates where the vector is pointing and can vary significantly depending on its components. In the context of the unit vector, which has a magnitude of 1, the direction remains consistent with the original vector from which it is derived. By dividing the original vector by its magnitude, we preserve the direction while standardizing the magnitude to 1. With our one-dimensional vector example, \( \mathbf{w} = -3 \mathbf{i} \), when we calculate its unit vector, \( \mathbf{u}_w = -\mathbf{i} \), we observe that its direction is along the negative x-axis. Thus, by scaling the original vector to become a unit vector, we maintain its direction but adjust its magnitude to equal 1, ensuring it remains streamlined yet informative about its original orientation.

One-Dimensional Vector

A one-dimensional vector is the simplest form of a vector, having only one component. This type of vector lies along a single axis in a vector space, making it intuitive to understand and utilize. In practical terms, a one-dimensional vector like \( -3 \mathbf{i} \) exists entirely along the x-axis. This simplicity means calculating properties such as magnitude or direction typically involves fewer steps compared to vectors in higher-dimensional spaces. Using this one-dimensional approach is especially useful when examining basic linear motion or transformations, where the essence of the problem revolves around moving along a straight path. Consequently, one-dimensional vectors serve as the building blocks for more complicated vector operations encountered in multidimensional spaces.