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

Name: Problem 10, Chapter 10: Algebra and Trigonometry
Uploaded: 2020-06-17T12:27:34-08:00
Duration: 18 s
Channel: Ashley Hanson
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 The answer to this question is D. To find vector V, we need mod V multiplied by sine alpha i plu

Question

If 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 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) $|\mathbf{v}|(\sin \alpha \mathbf{i}+\cos \alpha \mathbf{j})$

Algebra and Trigonometry

Michael Sullivan 10th Edition

Chapter 10, Problem 10

Instant Answer

Solved by Expert Ashley Hanson

06/17/2020

Step 1

Step 1: First, we need to understand that the direction angle $\alpha$ is the angle between the vector $\mathbf{v}$ and the unit vector $\mathbf{i}$ along the x-axis. Show more…

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If 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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