If 𝐯 is a nonzero vector with direction angle α, 0^∘ ≤α<360^∘, between 𝐯 and 𝐢, then 𝐯 equals wh

Name: Problem 10, Chapter 8: Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry
Uploaded: 2021-04-16T06:08:23-08:00
Duration: 2 min 4 s
Channel: Suman Saurav Thakur
Description: If 𝐯 is a nonzero vector with direction angle α, 0^∘ ≤α<360^∘, between 𝐯 and 𝐢, then 𝐯 equals which of the following? (a) 𝐯(cosα𝐢-sinα𝐣) (b) 𝐯(cosα𝐢+sinα𝐣) (c) 𝐯(sinα𝐢-cosα𝐣) (d) 𝐯(sinα𝐢+cosα𝐣)

step 1 we need to just write the vector in component form okay so it is given that they have a vector v

Question

If $\mathbf{v}$ is a nonzero vector with direction angle $\alpha, 0^{\circ} \leq \alpha<360^{\circ}$, between $\mathbf{v}$ and $\mathbf{i}$, then $\mathbf{v}$ equals which of the following? (a) $\|\mathbf{v}\|(\cos \alpha \mathbf{i}-\sin \alpha \mathbf{j})$ (b) $\|\mathbf{v}\|(\cos \alpha \mathbf{i}+\sin \alpha \mathbf{j})$ (c) $\|\mathbf{v}\|(\sin \alpha \mathbf{i}-\cos \alpha \mathbf{j})$ (d) $\|\mathbf{v}\|(\sin \alpha \mathbf{i}+\cos \alpha \mathbf{j})$

    If $\mathbf{v}$ is a nonzero vector with direction angle $\alpha, 0^{\circ} \leq \alpha<360^{\circ}$, between $\mathbf{v}$ and $\mathbf{i}$, then $\mathbf{v}$ equals which of the following?
(a) $\|\mathbf{v}\|(\cos \alpha \mathbf{i}-\sin \alpha \mathbf{j})$
(b) $\|\mathbf{v}\|(\cos \alpha \mathbf{i}+\sin \alpha \mathbf{j})$
(c) $\|\mathbf{v}\|(\sin \alpha \mathbf{i}-\cos \alpha \mathbf{j})$
(d) $\|\mathbf{v}\|(\sin \alpha \mathbf{i}+\cos \alpha \mathbf{j})$

Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry

Michael Sullivan 4th Edition

Chapter 8, Problem 10

Instant Answer

Solved by Expert Suman Saurav Thakur

04/16/2021

Step 1

We are given a nonzero vector $\mathbf{v}$ with a direction angle $\alpha$ between $\mathbf{v}$ and the positive x-axis (represented by $\mathbf{i}$). We need to express $\mathbf{v}$ in terms of its magnitude and direction. Show more…

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If $\mathbf{v}$ is a nonzero vector with direction angle $\alpha, 0^{\circ} \leq \alpha<360^{\circ}$, between $\mathbf{v}$ and $\mathbf{i}$, then $\mathbf{v}$ equals which of the following? (a) $\|\mathbf{v}\|(\cos \alpha \mathbf{i}-\sin \alpha \mathbf{j})$ (b) $\|\mathbf{v}\|(\cos \alpha \mathbf{i}+\sin \alpha \mathbf{j})$ (c) $\|\mathbf{v}\|(\sin \alpha \mathbf{i}-\cos \alpha \mathbf{j})$ (d) $\|\mathbf{v}\|(\sin \alpha \mathbf{i}+\cos \alpha \mathbf{j})$

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Key Concepts

Magnitude and Direction

A vector in the plane is characterized by both its magnitude (or length) and its direction. The magnitude represents the distance from the origin to the point described by the vector, while the direction is given by an angle measured from a fixed axis, often the positive x-axis.

Component Form of a Vector

Vectors can be expressed in terms of their horizontal and vertical components. By using trigonometric functions, the horizontal component is obtained by multiplying the magnitude by the cosine of the direction angle, and the vertical component by the sine of that angle. This leads to the representation: v = |v|(cos(?)i + sin(?)j), which clearly separates the contributions in the i (horizontal) and j (vertical) directions.

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