Consider the vector v with rectangular components (a, b) and polar components (r, θ).
- What is the relationship between rand and the horizontal component, a?
Write a formula a = - What is the relationship between rand and the vertical component, b?
Write a formula b =
What is the relationship between a and b and the magnitude, r?
Write a formula r =
3. - What is the relationship between a and b and the angle, θ?
Write an e - quation relating these three values.
The Correct Answer and Explanation is :
1. Relationship between ( r ) and the horizontal component ( a ):
The horizontal component ( a ) can be expressed in terms of the polar components as follows:
[
a = r \cos(\theta)
]
Explanation:
In polar coordinates, ( r ) represents the magnitude (or length) of the vector, and ( \theta ) represents the angle the vector makes with the positive x-axis. The horizontal component of the vector, ( a ), is found by multiplying the magnitude ( r ) by the cosine of the angle ( \theta ). This uses basic trigonometry, as the cosine function gives the adjacent side of a right triangle in a unit circle.
2. Relationship between ( r ) and the vertical component ( b ):
The vertical component ( b ) is given by:
[
b = r \sin(\theta)
]
Explanation:
Similarly to the horizontal component, the vertical component is the projection of the vector onto the y-axis. The sine function gives the ratio of the opposite side in a right triangle, so multiplying the magnitude ( r ) by ( \sin(\theta) ) gives the length of the vertical component ( b ). This is a direct application of trigonometry.
3. Relationship between ( a ), ( b ), and the magnitude ( r ):
The magnitude ( r ) is related to ( a ) and ( b ) by the Pythagorean theorem:
[
r = \sqrt{a^2 + b^2}
]
Explanation:
The magnitude of the vector is the hypotenuse of a right triangle whose sides are the components ( a ) (horizontal) and ( b ) (vertical). By the Pythagorean theorem, the magnitude ( r ) is the square root of the sum of the squares of ( a ) and ( b ).
4. Relationship between ( a ), ( b ), and the angle ( \theta ):
The angle ( \theta ) is given by:
[
\theta = \tan^{-1}\left(\frac{b}{a}\right)
]
Explanation:
The angle ( \theta ) can be found using the arctangent function, which is the inverse of the tangent. The tangent of an angle in a right triangle is the ratio of the opposite side (vertical component ( b )) to the adjacent side (horizontal component ( a )). Therefore, ( \theta ) can be calculated as the arctangent of ( b/a ), which gives the angle the vector makes with the positive x-axis.