Question

Show, using the definition of the Cross Product, that \(\vec{u} \times \vec{u}=\) \(\overrightarrow{0}\)

Short answer

\(\vec{u} \times \vec{u} = \overrightarrow{0}\) because the cross product of any vector with itself is zero.

Step-by-step solution

Define Cross Product

The cross product of two vectors \(\vec{a} = \langle a_1, a_2, a_3 \rangle\) and \(\vec{b} = \langle b_1, b_2, b_3 \rangle\) is a vector \(\vec{c} = \vec{a} \times \vec{b}\) given by: \[\vec{c} = \langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rangle \]

Set Up the Cross Product for \(\vec{u} \times \vec{u}\)

Let \(\vec{u} = \langle u_1, u_2, u_3 \rangle\). When calculating the cross product \(\vec{u} \times \vec{u}\), substitute \(\vec{u}\) for both vectors in the formula: \[\vec{u} \times \vec{u} = \langle u_2u_3 - u_3u_2, u_3u_1 - u_1u_3, u_1u_2 - u_2u_1 \rangle \]

Simplify the Expression

When simplifying the components from the last step, note that each component results in zero: 1) \(u_2u_3 - u_3u_2 = 0\), 2) \(u_3u_1 - u_1u_3 = 0\), 3) \(u_1u_2 - u_2u_1 = 0\). Thus, \(\vec{u} \times \vec{u} = \langle 0, 0, 0 \rangle\).

State the Conclusion

By definition, the result of the above operations gives the zero vector: \(\vec{u} \times \vec{u} = \overrightarrow{0}\). This confirms that the cross product of any vector with itself is the zero vector.

Key concepts

Vector Algebra

Vector algebra is a branch of mathematics that deals with vectors and the operations that can be performed on them. Vectors are entities that have both magnitude (size) and direction. They are often represented as arrows in space or as an array of numbers in a coordinate system. In vector algebra, you will encounter several key operations, including addition, subtraction, scalar multiplication, dot products, and cross products.
Cross product, a major focus in vector algebra, involves multiplying two vectors to produce a third vector. This product reveals many important physical quantities, such as torque in physics. Additionally, the cross product is fundamentally different from the dot product, which results in a scalar. Knowing when to use each operation is crucial for solving problems in both mathematics and physics.
Understanding vector algebra is essential because it lays the foundation for areas such as physics, engineering, and computer graphics. It helps describe movement, force, and the properties of three-dimensional space efficiently.

Zero Vector

The zero vector is an essential concept in vector algebra. It represents a vector with no magnitude and no specific direction. Mathematically, this vector is expressed as \(\overrightarrow{0}\), which can also be written as \(\langle 0, 0, 0 \rangle\) in three-dimensional space.
The zero vector acts as the identity element in vector addition, meaning that when any vector is added to the zero vector, the result is the original vector itself. Similarly, when a vector is subtracted from itself, the result is the zero vector.
In the context of cross products, the zero vector is particularly important. If two vectors are parallel or identical, their cross product results in the zero vector. For example, in the exercise above, the cross product of a vector with itself yields the zero vector, reinforcing the concept that no direction or magnitude change can occur when a vector interacts with itself in this manner.

Vector Operations

Vector operations are the procedures used to manipulate vectors in mathematics and physics. These operations are crucial for solving complex problems and understanding the physical world. Some key vector operations include vector addition, subtraction, scalar multiplication, and both dot and cross products.
Let's explore the cross product in more detail, as it's a central operation in vector analysis. The cross product of two vectors \(\vec{a}\) and \(\vec{b}\) results in another vector that is perpendicular to the plane containing \(\vec{a}\) and \(\vec{b}\). This vector can be calculated using the formula \[\vec{c} = \langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rangle\]. Notice the result relies heavily on determinants and algebraic expansion.
Besides cross products, understanding vector operations means knowing how to add vectors by summing their respective components and subtracting by taking their differences. Scalar multiplication involves multiplying a vector by a number, which changes its magnitude but not its direction.

• Addition: Combines vectors end-to-end.
• Subtraction: Finds the difference between two vectors.
• Scalar Multiplication: Alters vector size.
• Cross Product: Computes a perpendicular vector.
• Dot Product: Produces a scalar, showing magnitude alignment.

By understanding these operations, you’ll be better equipped to handle a wide variety of mathematical and engineering challenges.