A vector with 𝐯=1 is known as a unit vector. Prove that if 𝐯, 𝐰 are both unit vectors, then 𝐯+𝐰

step 1 Because U and V are unit vectors, the magnitude of U is one and the magnitude of V is one. Becau

A vector with 𝐯=1 is known as a unit vector. Prove that if...

A vector with $\|\mathbf{v}\|=1$ is known as a unit vector. Prove that if $\mathbf{v}, \mathbf{w}$ are both unit vectors, then $\mathbf{v}+\mathbf{w}$ and $\mathbf{v}-\mathbf{w}$ are orthogonal. Are they also unit vectors?

Key Concepts

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number. It is defined as the sum of the products of corresponding entries and encodes geometric information such as the angle between two vectors. In particular, if the dot product is zero, the vectors are orthogonal.

Unit Vectors

A unit vector is one with a norm (or magnitude) equal to 1. They are useful in representing directions without regard to length, and frequently appear in proofs and calculations where normalization of vectors is required.

Orthogonality

Orthogonality in a vector space means that two vectors are perpendicular to each other. This property can be verified by checking that their dot product is zero, which indicates that there is no component of one vector in the direction of the other.

Vector Addition

Vector addition is an operation that combines two vectors to produce another vector. The geometrical interpretation is the parallelogram or triangle rule, where the resultant vector represents the diagonal of the parallelogram formed by the two original vectors.

Vector Subtraction

Vector subtraction is defined as the addition of one vector with the negative of another vector. Geometrically, subtracting a vector can be interpreted as reversing its direction and then adding it to the other vector, which is useful in understanding relative positions or differences between vectors.

Vector Norm

The vector norm is a function that assigns a positive length or magnitude to each vector in a vector space. Commonly used norms include the Euclidean norm, which is defined as the square root of the sum of the squares of the vector's components. The norm is central to the concepts of unit vectors and helps to determine distances and angles in the vector space.

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