Problem 27 - Thomas Calculus

Question

Express each vector as a product of its length and direction. 5 \(\mathbf{k}\)

Step-by-Step Solution

Answer

5\(\mathbf{k}\) is the product of 5 (length) and \(\mathbf{k}\) (direction).

Step 1: Identify the Components of the Vector

Given the vector expression is 5 \( \mathbf{k} \). Here, \( \mathbf{k} \) is a unit vector in the positive z-direction. A unit vector has a magnitude of 1.

Step 2: Calculate the Length of the Vector

The scalar 5 represents the length (or magnitude) of the vector. So, the length of the vector is 5.

Step 3: Determine the Direction Vector

The direction of the vector is given by the unit vector \( \mathbf{k} \). Since \( \mathbf{k} \) is a unit vector along the z-axis, it already represents the direction.

Step 4: Express the Vector as a Product

The vector can be expressed as the product of its length and its direction. Therefore, 5 \( \mathbf{k} \) is equal to the product of its magnitude, 5, and its direction, \( \mathbf{k} \). Mathematically, it's represented as: \[ 5 \mathbf{k} = 5 \times \mathbf{k} \].

Key Concepts

Unit Vector

A unit vector is a vector that has a magnitude of 1. It is often used to indicate direction without influencing magnitude. Think of it as a compass that shows which direction to move in, but not how far or fast to go. In any space, unit vectors are essential in direction representation.

• They are denoted by placing a hat symbol (^) above the vector symbol, although sometimes simply labeling it as a common unit vector like \( \mathbf{i} \), \( \mathbf{j} \), or \( \mathbf{k} \) in three-dimensional Cartesian coordinates.
• A key feature of unit vectors is their ability to form the basis for any vector in space when combined with scalar multiplication, representing both direction and proportional components.
• They are crucial in simplifying complex problems by separating the direction from the magnitude.

Understanding unit vectors helps in visualizing how any vector's direction can be highlighted independently from its length.

Vector Magnitude

The magnitude of a vector is a crucial attribute that measures the length or size of the vector. It tells you how "long" the vector is and gives a scalar value.

• To find the magnitude of a vector \( \mathbf{v} = \langle a, b, c \rangle \) in three-dimensional space, use the formula: \[ |\mathbf{v}| = \sqrt{a^2 + b^2 + c^2} \]
• Magnitude is always non-negative, because it represents a measurement that cannot be less than zero.
• Vector magnitude helps quantify the vector's effect in applications like physics and engineering, where both direction and intensity matter.

Magnitude is often accompanied by a unit vector to form the complete representation of a vector, separating the vector's size from its direction. This was illustrated in the given exercise where the magnitude of 5 represented the strength of the vector \( 5 \mathbf{k} \).

Direction Vector

The direction vector uniquely specifies the orientation of a vector in space. It helps in understanding where and how a vector extends.

• A direction vector can be simply a unit vector, like \( \mathbf{k} \), whose main purpose is to show the direction the vector is pointing towards.
• By having a direction vector, we can express any vector as the product of its magnitude and its direction vector, making calculations intuitive.
• It is an effective tool in navigation, physics, and various fields requiring movement or shifts in specific paths.

This concept was used in our exercise where \( \mathbf{k} \) served as the direction vector. It confirmed that \( 5 \mathbf{k} \), being directed along the z-axis, utilizes this unit vector's orientation, along with its magnitude of 5, to define the complete vector.