- A vector in the x-y plane has a length of 4.5 m and a y component of -2.7 m. What angle does this vector make with the x axis? Why does this problem have two answers?
- Consider a map oriented so that the x axis runs east-west (with east being the “positive” direction) and y runs north-south (with north “positive”). A person drives 15 km to the west, then turns and drives 45 km to the south. Find the x and y components of the total movement of the person.
The Correct Answer and Explanation is :
Problem 1: Angle of the Vector in the x-y Plane
Given:
- Length of the vector = 4.5 m
- y-component of the vector = -2.7 m
We can use the Pythagorean theorem to find the x-component of the vector. The length of the vector, ( \text{Magnitude of vector} ), is given by:
[
\text{Magnitude} = \sqrt{(x^2 + y^2)}
]
Substituting the known values:
[
4.5 = \sqrt{(x^2 + (-2.7)^2)}
]
Squaring both sides:
[
4.5^2 = x^2 + 2.7^2
]
[
20.25 = x^2 + 7.29
]
Now, solving for ( x^2 ):
[
x^2 = 20.25 – 7.29 = 12.96
]
So, the x-component of the vector is:
[
x = \sqrt{12.96} = 3.6 \text{ m}
]
Now, the angle ( \theta ) that the vector makes with the positive x-axis can be found using the trigonometric relationship:
[
\tan(\theta) = \frac{\text{y-component}}{\text{x-component}} = \frac{-2.7}{3.6}
]
[
\theta = \tan^{-1}\left(\frac{-2.7}{3.6}\right)
]
Using a calculator:
[
\theta \approx \tan^{-1}(-0.75) \approx -36.87^\circ
]
Two Possible Answers:
The reason there are two possible answers is due to the fact that the tangent function has a periodicity of ( 180^\circ ). So, the vector could be in the fourth quadrant (as calculated above, ( -36.87^\circ )) or in the second quadrant (where the x-component is negative and y-component is negative, which would be ( 180^\circ – 36.87^\circ = 143.13^\circ )).
Problem 2: Finding the Components of the Person’s Movement
Given:
- First movement: 15 km west (negative x-direction)
- Second movement: 45 km south (negative y-direction)
The components of the total movement are simply the sums of the individual components in the x and y directions.
- Movement to the West:
- x-component: ( -15 ) km (since west is negative in the x-direction)
- y-component: ( 0 ) km (no movement in the y-direction)
- Movement to the South:
- x-component: ( 0 ) km (no movement in the x-direction)
- y-component: ( -45 ) km (since south is negative in the y-direction)
Now, adding these components:
- Total x-component: ( -15 + 0 = -15 ) km
- Total y-component: ( 0 + (-45) = -45 ) km
Thus, the total movement of the person has the following components:
- x-component = ( -15 ) km (west)
- y-component = ( -45 ) km (south)
Explanation:
The components represent how far the person has moved in each direction. The negative sign indicates that the person has moved to the west and south, respectively. The total movement can be represented as a vector whose components indicate displacement in the horizontal (x) and vertical (y) directions. By resolving the movement into components, we can better understand the overall direction and magnitude of the displacement.
I’ll also generate an image illustrating these two problems.
Here is the diagram illustrating the two physics problems. It shows the vector components and the person’s movement on the map, along with labeled x and y components for both problems. Let me know if you need any further explanation or adjustments!