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Chapter 3: Problem 15

Use unit vectors to express a displacement of \(120 \mathrm{km}\) at \(29^{\circ}\) counterclockwise from the \(x\) -axis.

Short Answer

Expert verified
The displacement can be described as \( \vec{d} = 120 \cos(29^{\circ}) \hat{i} + 120 \sin(29^{\circ}) \hat{j} \)

Step by step solution

01

Determine Unit Vectors

In two-dimensional space, a unit vector along the \(x\)-axis is denoted as \(\hat{i}\) and along the \(y\)-axis as \(\hat{j}\). These unit vectors have magnitude of 1 and the direction of positive \(x\) and \(y\) axis respectively.
02

Denote Displacement Vector

The displacement vector of 120 km at an angle of \(29^{\circ}\) can be represented in terms of the \(x\) and \(y\) unit vectors as follows: \( \vec{d} = d_x \hat{i} + d_y \hat{j} \). Here, \(d_x\) and \(d_y\) are the components of the displacement vector in the \(x\)-axis and \(y\)-axis directions respectively.
03

Compute the x and y Components

The components are determined by: \(d_x = d \cos(\alpha)\) and \(d_y = d \sin(\alpha)\). Here, \(d\) is the magnitude of the displacement (120 km) and \(\alpha\) is the angle of the displacement with respect to the \(x\)-axis (29 degrees). Therefore, \(d_x = 120 \cos(29^{\circ})\), \(d_y = 120 \sin(29^{\circ})\).
04

Express Displacement Vector in Terms of Unit Vectors

The displacement vector is then expressed as \( \vec{d} = 120 \cos(29^{\circ}) \hat{i} + 120 \sin(29^{\circ}) \hat{j} \)

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