Problem 19 - Algebra and Trigonometry with Analytic Geometry

Question

\(\mathbf{a}+(-\mathbf{a})=0 \quad \)

Step-by-Step Solution

Answer

The sum of a vector and its negative is always the zero vector.

Step 1: Identify the Vector and Its Negative

Here, we are dealing with a vector \( \mathbf{a} \), and its negative is \( -\mathbf{a} \). To find \( -\mathbf{a} \), you simply take the vector \( \mathbf{a} \) and reverse its direction where each component of \( \mathbf{a} \) is replaced by its negative.

Step 2: Apply Vector Addition

Now, we add the original vector \( \mathbf{a} \) and its negative \( -\mathbf{a} \). In vector addition, you add corresponding components of the vectors: each component of \( \mathbf{a} \) is added to its corresponding component in \( -\mathbf{a} \).

Step 3: Verify the Result

For each component, if you add a number and its negative, their sum is zero. Hence, each component sum of the vector \( \mathbf{a} \) and \( -\mathbf{a} \) is zero, and therefore their entire sum is the zero vector, written as \( 0 \) or \( \mathbf{0} \). This is because every point in the vector has its equivalent negative, causing them to cancel out.

Key Concepts

Negative Vector

A negative vector is simply a vector that has the same magnitude but the opposite direction to another vector. To find the negative of a vector, you take each component of the original vector and multiply it by -1. For example, if you have a vector \( \mathbf{a} = (x, y) \), its negative, denoted as \( -\mathbf{a} \), would be \( (-x, -y) \).
In graphical terms, imagine the vector \( \mathbf{a} \) pointing in one direction on the Cartesian coordinate system. The vector \( -\mathbf{a} \) will point directly opposite to it. This reversal of direction while maintaining the same length is the essence of a negative vector.
Bullets of reversal characteristics:

• The vector's length (or magnitude) remains unchanged.
• The vector's direction is reversed.
• Each component of the vector flips sign.

These properties are crucial when performing operations like addition or subtraction, as they help determine the overall direction and resulting vector.

Zero Vector

The zero vector, often denoted as \( \mathbf{0} \), is a special vector that has all of its components equal to zero. It essentially represents the idea of 'no movement' or 'no change' in a multi-dimensional space. Graphically, it can be depicted as a single point at the origin of the axes, since it has no direction or magnitude.
In mathematical operations, when you add a vector to its negative, as shown in the equation \( \mathbf{a} + (-\mathbf{a}) = \mathbf{0} \), their sum is the zero vector. Here is why:

• Adding corresponding components of a vector and its negative results in zero for each component.
• For example, if \( \mathbf{a} = (x, y) \), then \( \mathbf{a} + (-\mathbf{a}) = (x + (-x), y + (-y)) = (0, 0) \).
• This zero vector result indicates no resultant movement or change, as both vectors perfectly cancel each other out.

Understanding the zero vector is pivotal because it acts as the additive identity in vector mathematics, similar to how the number 0 is the additive identity for real numbers.

Vector Components

Vector components are the building blocks of a vector in multi-dimensional space. Any vector can be broken down into components that show its influence in each dimension. For instance, in a 2D vector \( \mathbf{a} = (a_x, a_y) \), \( a_x \) and \( a_y \) are the vector's components in the x and y directions, respectively.
Components allow us to represent the vector’s impact in each direction individually, which is very useful in operations like vector addition, subtraction, and more.
Key points about vector components:

• They allow for easy computation of vector operations by adding or subtracting corresponding components.
• Vectors can be reconstructed from their components using the formula \( \mathbf{a} = (a_x, a_y) \) in 2D, \( (a_x, a_y, a_z) \) in 3D, etc.
• Understanding components helps visualize vector interactions in a coordinate system.

Whether dealing with simple physics problems or complex engineering scenarios, breaking vectors into components simplifies calculations and provides clear insights into how vectors behave.