In Problems 59-64, write the vector 𝐯 in the form a +b 𝐣, given its magnitude 𝐯 and the angle αi

Name: Problem 59, Chapter 8: Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry
Uploaded: 2020-07-04T19:17:57-08:00
Duration: 51 s
Channel: Rakesh Kumar Sharma
Description: In Problems 59-64, write the vector 𝐯 in the form a +b 𝐣, given its magnitude 𝐯 and the angle αit makes with the positive x-axis. 𝐯=5, α=60^∘

step 1 In this question, the models of Vector V is given as 5 and the angle it makes with the positive

Question

In Problems 59-64, write the vector $\mathbf{v}$ in the form a $+b \mathbf{j}$, given its magnitude $\|\mathbf{v}\|$ and the angle $\alpha$ it makes with the positive $x$-axis. $\|\mathbf{v}\|=5, \quad \alpha=60^{\circ}$

   In Problems 59-64, write the vector $\mathbf{v}$ in the form a $+b \mathbf{j}$, given its magnitude $\|\mathbf{v}\|$ and the angle $\alpha$ it makes with the positive $x$-axis.
$\|\mathbf{v}\|=5, \quad \alpha=60^{\circ}$

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

Michael Sullivan 4th Edition

Chapter 8, Problem 59

Instant Answer

Solved by Expert Rakesh Kumar Sharma

07/04/2020

Step 1

The $x$-component of $\mathbf{v}$ can be found using the formula $v_x = \|\mathbf{v}\| \cos \alpha$, so in this case, $v_x = 5 \cos 60^\circ = \frac{5}{2}$. The $y$-component of $\mathbf{v}$ can be found using the formula $v_y = \|\mathbf{v}\| \sin \alpha$, so Show more…

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In Problems 59-64, write the vector $\mathbf{v}$ in the form a $+b \mathbf{j}$, given its magnitude $\|\mathbf{v}\|$ and the angle $\alpha$ it makes with the positive $x$-axis. $\|\mathbf{v}\|=5, \quad \alpha=60^{\circ}$

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Key Concepts

Component Form of Vectors

This concept involves expressing a vector as the sum of its horizontal and vertical parts. It allows one to write any vector in two dimensions as a linear combination of unit vectors along the x-axis and y-axis, making computations and interpretations more straightforward.

Trigonometric Decomposition

This is the process of breaking down a vector into its components using trigonometric functions. By using the cosine function for the horizontal component and the sine function for the vertical component, one can relate a vector's magnitude and its direction (specified by an angle) to its components.

Standard Basis Vectors

These are the unit vectors typically denoted by i and j in two-dimensional Cartesian coordinates. They represent the directions along the x-axis and y-axis respectively, and serve as the building blocks for expressing vectors in component form.

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