Problem 69 - Algebra and Trigonometry

Question

What are orthogonal vectors?

Step-by-Step Solution

Answer

Orthogonal vectors are vectors that are perpendicular to each other, which in a Cartesian coordinate system implies their dot product will be zero.

Step 1: Defining Orthogonal Vectors

Orthogonal Vectors are two or more vectors that are perpendicular to each other. This means that they meet at a right angle (90 degrees). In a Cartesian coordinate system, this implies that their dot product will be zero.

Step 2: Understanding Dot Product

The Dot Product also known as the scalar product, of two vectors is the operation which combines two vectors to form a scalar. It is represented as:\( \vec{A} . \vec{B} = |\vec{A}||\vec{B}|Cosθ \) where θ is the angle between \(\vec{A}\) and \(\vec{B}\). In the case of orthogonal vectors since the angle between them is 90 degrees, Cos 90 = 0. Therefore their Dot product becomes 0.

Step 3: Practical Implication

So, given vectors are orthogonal when the dot product equals to zero. You can check if two vectors are orthogonal by calculating their dot product, if it equals zero, they are orthogonal. This property is often used to simplify calculations in physics and engineering.