If 𝐯 is a unit vector and the angle between 𝐯 and 𝐢 is α, show that 𝐯=cosα𝐢+sinα𝐣.

Name: Problem 40, Chapter 8: Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry
Uploaded: 2020-06-19T08:56:17-08:00
Duration: 59 s
Channel: Jodi Folley
Description: If 𝐯 is a unit vector and the angle between 𝐯 and 𝐢 is α, show that 𝐯=cosα𝐢+sinα𝐣.

step 1 So we have two vectors, V and I, and V is a unit vector, which means its length is 1. So I'm goi

If 𝐯 is a unit vector and the angle between 𝐯 and 𝐢 is α,...

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Key Concepts

Trigonometric Representation of Vectors

A vector in the plane can be expressed in terms of its angle with respect to a coordinate axis by decomposing it into cosine and sine components. Specifically, if a vector forms an angle with a basis axis, its magnitude multiplied by the cosine of the angle gives the component along that axis, while the sine gives the component perpendicular to it. When the vector is a unit vector, these components become exactly the cosine and sine of the angle.

Unit Vector

A unit vector is a vector with a magnitude of one. It is used to indicate direction and can be scaled to represent any other vector by multiplying it with the desired magnitude. In problems involving angles, unit vectors simplify calculations since they focus solely on direction.

Standard Basis Vectors

Standard basis vectors, often denoted as i and j in two-dimensional space, form the fundamental set of orthogonal directions along the axes. They allow any vector in the plane to be expressed as a linear combination of these two vectors, making it easier to work with vector components.

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