Question

T/F: A fundamental principle of the cross product is that \(\vec{u} \times \vec{v}\) is orthogonal to \(\vec{u}\) and \(\vec{v} .\)

Short answer

True, \(\vec{u} \times \vec{v}\) is orthogonal to \(\vec{u}\) and \(\vec{v}\).

Step-by-step solution

Understanding Cross Product

When calculating the cross product of two vectors, \(\vec{u} \times \vec{v}\), the resulting vector is perpendicular to the plane formed by the original vectors \(\vec{u}\) and \(\vec{v}\). This orthogonal property is a key principle of the cross product.

Checking Orthogonality Condition

The condition of orthogonality states that if \(\vec{w} = \vec{u} \times \vec{v}\), then \(\vec{w}\) should satisfy the dot product condition: \(\vec{u} \cdot \vec{w} = 0\) and \(\vec{v} \cdot \vec{w} = 0\). These conditions confirm that \(\vec{w}\) is orthogonal to both \(\vec{u}\) and \(\vec{v}\) because the dot product of orthogonal vectors is zero.

Conclusion

Since the cross product \(\vec{u} \times \vec{v}\) is indeed orthogonal to both \(\vec{u}\) and \(\vec{v}\), the statement given in the exercise is true.

Key concepts

Orthogonal Vectors

Orthogonal vectors are two or more vectors that are at right angles, or perpendicular, to each other. This perpendicular relationship is a fundamental aspect of geometry and vector mathematics.
To determine if two vectors are orthogonal, we often use their dot product. If two vectors, say \( \vec{a} \) and \( \vec{b} \), are orthogonal, then their dot product equals zero: \( \vec{a} \cdot \vec{b} = 0 \). This zero value is characteristic of orthogonal vectors.
If you imagine two arrows in space, orthogonal vectors would resemble two arrows meeting at a right angle.

• This concept applies in physics, for instance, when calculating forces that act perpendicularly.
• In graphics, it helps in determining visibility or shading by checking if a light ray is orthogonal to a surface.

Understanding orthogonality aids in various real-world applications, simplifying complex systems into manageable parts.

Dot Product

The dot product, also known as the scalar product, is an operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It conveys how much of one vector overlaps with another.
Mathematically, given two vectors \( \vec{a} = [a_1, a_2, a_3] \) and \( \vec{b} = [b_1, b_2, b_3] \), their dot product is calculated as \( \vec{a} \cdot \vec{b} = a_1 \cdot b_1 + a_2 \cdot b_2 + a_3 \cdot b_3 \).
Since it returns a scalar value, the dot product is also indicative of vector directions. If the dot product is zero, the vectors are orthogonal. Consequently, it shows the level of alignment between two vectors.

• Used to find angles: The cosine of the angle between two vectors involves the dot product, \( \cos(\theta) = \frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\| \|\vec{b}\|} \).
• Applications in physics include understanding work done, where force and displacement are vectors used in computations.

Though simple in concept, the dot product is very powerful in vector mathematics.

Vector Orthogonality

Vector orthogonality is a specific mathematical condition where two vectors meet at a right angle. This property is highly significant in various fields, including physics, engineering, and computer science.
Orthogonality possesses the unique characteristic that the dot product of two orthogonal vectors is zero, mathematically represented as \( \vec{u} \cdot \vec{v} = 0 \). In vector spaces, orthogonality simplifies the process of computations and clarifies geometric interpretations.
The orthogonality of vectors underpins the principle of the cross product, where the resultant vector is orthogonal to the original vectors. This concept confirms the statement in the original exercise that the cross product \( \vec{u} \times \vec{v} \) is orthogonal to both \( \vec{u} \) and \( \vec{v} \).

• Orthogonal vectors form the basis of orthonormal bases in vector spaces, providing simpler expressions for vector components.
• The absence of projection between orthogonal vectors is what leads to their dot product being zero.

By continually using orthogonality, calculations in 3D space become intuitive and significantly easier to manage.