Question

If \(\mathbf{v}\) is a vector, then \(\mathbf{v}+(-\mathbf{v})=\) _____

Short answer

\(\mathbf{v} + (-\mathbf{v}) = \mathbf{0}\)

Step-by-step solution

Identify the given vector

The problem states that \(\mathbf{v}\) is a vector. This means \(\mathbf{v}\) can represent any vector in a vector space. For example, \(\mathbf{v} = \langle x, y, z \rangle \) in three-dimensional space.

Understand the negative of a vector

The negative of a vector \(\mathbf{v}\), written as \(-\mathbf{v}\), is a vector of the same magnitude but in the opposite direction. If \(\mathbf{v} = \langle x, y, z \rangle\), then \(-\mathbf{v} = \langle -x, -y, -z \rangle\).

Apply vector addition

To find \(\mathbf{v} + (-\mathbf{v})\), add each corresponding component of the vectors together: \[ \mathbf{v} + (-\mathbf{v}) = \langle x, y, z \rangle + \langle -x, -y, -z \rangle \].

Perform the component-wise addition

Add the components: \[ \langle x + (-x), y + (-y), z + (-z) \rangle = \langle 0, 0, 0 \rangle \].

Conclusion

The result of adding a vector and its negative is always the zero vector. Thus, \(\mathbf{v} + (-\mathbf{v}) = \mathbf{0}\).

Key concepts

Vector Space

A vector space is a fundamental concept in linear algebra. It is a set of vectors that can be added together and multiplied by scalars to produce another vector within the same set. Vectors in a vector space can have one or more dimensions, like \(\begin{aligned} \textbf{v} = \begin{pmatrix} x \ y \ z \end{pmatrix} \text{ in a three-dimensional space.} \end{aligned}\). The properties of a vector space include closure under addition and scalar multiplication, the existence of a zero vector, and more. Each combination of these operations must follow specific rules, called axioms, that govern their behavior. These rules ensure that any manipulation follows a predictable pattern.

Negative Vector

A negative vector, denoted as \(-\textbf{v}\), is a vector that has the same magnitude as the original vector \(\textbf{v}\), but points in the opposite direction. This is achieved by changing the sign of each component of the vector. For example, if \(\textbf{v} = \begin{pmatrix} x \ y \ z \ \end{pmatrix}\), then \(-\textbf{v} = \begin{pmatrix} -x \ -y \ -z \ \end{pmatrix}\). The negative vector is crucial for various algebraic operations and helps us understand balancing forces in physics by providing a way to nullify vectors through addition.

Zero Vector

The zero vector, often written as \(\textbf{0}\), is a special vector with all components equal to zero. In any dimensional space, it is represented as \(\begin{pmatrix} 0 \ 0 \ 0 \end{pmatrix}\) in three dimensions, for example. The zero vector serves as the additive identity in a vector space, meaning that any vector added to the zero vector results in the original vector. Mathematically: \(\textbf{v} + \textbf{0} = \textbf{v}\). It is central to solving equations and understanding linear dependence and independence.

Component-Wise Addition

Component-wise addition is a method used to add two vectors. Each corresponding component of the vectors is added together to form a new vector. For vectors \(\textbf{v}\) and \(\textbf{u}\) in three-dimensional space, we add their coordinates like this: \(\textbf{v} + \textbf{u} = \begin{pmatrix} v_1 \ u_1 \end{pmatrix} + \begin{pmatrix} v_2 \ u_2 \end{pmatrix} + \begin{pmatrix} v_3 \ u_3 \end{pmatrix} = \begin{pmatrix} v_1 + u_1 \ v_2 + u_2 \ v_3 + u_3 \ \end{pmatrix}\). This straightforward method helps keep vector operations simple and visual. It is widely used in applications like physics and engineering.