Show that if 𝐮 and 𝐯 are orthogonal, then 𝐮 ×𝐯=𝐮𝐯

Name: Problem 58, Chapter 8: Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry
Uploaded: 2021-08-04T17:02:54-08:00
Duration: 1 min 14 s
Channel: Babita Kumari
Description: Show that if 𝐮 and 𝐯 are orthogonal, then 𝐮 ×𝐯=𝐮𝐯

Key Concepts

Vector Norm

The norm of a vector, often interpreted as its length or magnitude, is a fundamental concept in linear algebra. Norms provide a measure of the size of vectors, and they are crucial in quantifying distances and angles, thereby supporting various geometric and analytical computations in vector spaces.

Trigonometric Relationship in Vector Products

In the context of the cross product, the sine function plays a critical role by relating the magnitude of the cross product to the angle between the two vectors. When the vectors are orthogonal, the sine of the angle (which is 90°) becomes 1, leading to a direct equivalence between the product of the vectors’ norms and the magnitude of the cross product.

Orthogonality

Orthogonality in vector spaces signifies that the inner product (or dot product) of two vectors is zero. This property indicates that the vectors are perpendicular to each other, an important condition that often simplifies calculations in problems involving vector decomposition and projections.

Cross Product

The cross product is a binary operation on two vectors in three-dimensional space that yields a vector perpendicular to both original vectors. Its magnitude is given by the product of the magnitudes of the vectors and the sine of the angle between them, capturing both the area of the parallelogram spanned by the vectors and the notion of perpendicularity in three-dimensional space.

Transcript

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