Find two vectors in opposite directions that are orthogonal to the vector 𝐮. (The answers are no

step 1 What does it mean for us to find two vectors in opposite directions that are orthogonal to the v

Find two vectors in opposite directions that are orthogonal...

Key Concepts

Vector Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It is defined such that two vectors are orthogonal if and only if their dot product is zero, which is the basis for determining whether vectors are perpendicular to each other.

Orthogonal Vectors

Orthogonal vectors are vectors that meet at a right angle (90°) to each other. In many problems, finding vectors orthogonal to a given vector involves setting up the equation where the dot product equals zero and then choosing values for the free variables to generate specific solutions.

Antiparallel Vectors

Antiparallel vectors are vectors that have the same line of action but point in opposite directions. This means that one vector is a scalar multiple of the other by a negative number. Recognizing this property is useful when a problem asks for two vectors 'in opposite directions', as one can simply use any vector and its negative.

Free Variables and Under-determined Systems

When finding vectors orthogonal to a given vector, the condition from the dot product often leads to an under-determined system, meaning there are infinitely many solutions. In such cases, free variables can be assigned arbitrary values to generate a specific solution, and then their negatives yield the opposite direction counterpart.

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