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Chapter 3: Problem 5

Two displacement vectors each have magnitude \(20 \mathrm{km}\) One is directed \(60^{\circ}\) above the \(+x\) -axis; the other is directed \(60^{\circ}\) below the \(+x\) -axis. What is the vector sum of these two displacements? Use graph paper to find your answer.

Short Answer

Expert verified
Answer: To find the vector sum, first find the x and y components of each vector. Then, add the x-components and y-components separately. Finally, convert the sum of the components back into polar form (magnitude and angle) using the Pythagorean theorem and arctangent function. The magnitude of the vector sum and the angle of the vector sum will give the final answer.

Step by step solution

01

Find the components of the first vector

To find the x and y components of the first vector, we will use trigonometry. Recall that the x-component of a vector is given by its magnitude multiplied by the cosine of the angle it makes with the x-axis, while the y-component is given by its magnitude multiplied by the sine of that angle. For the first vector, magnitude = 20km and angle = 60 degrees above +x-axis. So, the x-component is 20*cos(60掳), and the y-component is 20*sin(60掳).
02

Find the components of the second vector

Similarly, for the second vector, magnitude = 20km and angle = 60 degrees below +x-axis. For this vector, we will take the angle as -60 degrees to account for the negative direction of the angle. The x-component is 20*cos(-60掳) and the y-component is 20*sin(-60掳).
03

Add the x-components and y-components of the two vectors

To find the vector sum, we will add the x-components and y-components separately. Sum of x-components = 20*cos(60掳) + 20*cos(-60掳) Sum of y-components = 20*sin(60掳) + 20*sin(-60掳)
04

Convert back to polar form

To convert back to polar form(pa and coord), we will use the Pythagorean theorem and the arctangent function. Magnitude of the vector sum = sqrt[(sum of x-components)^2 + (sum of y-components)^2] Angle of the vector sum = arctan(sum of y-components)/(sum of x-components) The magnitude of the vector sum and the angle of the vector sum will give the final answer.

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