Answer :
To convert a vector into unit vector notation, we first need to determine its x and y components based on its magnitude and the angle it makes with the positive x-axis. We then divide each component by the magnitude to get the unit vector notation.
Given a vector with a magnitude of 7.3 m and a direction of 250° counterclockwise from the positive x-direction, let's perform the calculations:
First, we convert the degree measure to radians because the trigonometric functions in most mathematical equations use radian measure:
[tex]\[ 250° = \left(\frac{250}{360}\right) \times 2\pi \approx 4.3633 \text{ radians} \][/tex]
(Here we're approximating the radian measure, typically you should keep it in terms of π or use a calculator to get an exact value).
The x and y components of a vector with an angle θ are found by:
[tex]\[ x = r \cos(\theta) \][/tex]
[tex]\[ y = r \sin(\theta) \][/tex]
Where:
- r is the magnitude of the vector,
- θ is the angle in radians measured counterclockwise from the positive x-axis.
Using the given magnitude r = 7.3 m and our converted angle θ = 4.3633 radians, we get the components:
[tex]\[ x = 7.3 \cos(4.3633) \][/tex]
[tex]\[ y = 7.3 \sin(4.3633) \][/tex]
Since the angle is more than 180° but less than 270°, we should expect both the x and y components to be negative (we are in the third quadrant of the coordinate system).
The x-component is A, which is the horizontal component, and here x would be:
[tex]\[ A = 7.3 \cos(4.3633) \approx -7.3 \times 0.342 \approx -2.4966 \][/tex]
(Here, I rounded the value of cos(250°), which is cos(4.3633 radians) to 0.342 for illustration purposes.)
The y-component is B, which is the vertical component, and here y would be:
[tex]\[ B = 7.3 \sin(4.3633) \approx -7.3 \times 0.9397 \approx -6.85981 \][/tex]
(Here, I rounded the value of sin(250°), which is sin(4.3633 radians) to 0.9397 for illustration purposes.)
To convert these components into unit vector notation, we will create a vector whose components are these values divided by the vector's magnitude. Since we're dealing with a unit vector, and its magnitude is 1, the components themselves represent the unit vector.
Unit vector notation:
[tex]\[ \vec{u} = \frac{1}{7.3}\vec{v} = \left(\frac{x}{7.3}, \frac{y}{7.3}\right) = \left(\frac{-2.4966}{7.3}, \frac{-6.85981}{7.3}\right) \][/tex]
Simplify the fractions to get the final unit vector notation:
[tex]\[ \vec{u} = (-0.342, -0.9397) \][/tex]
We could round these numbers as needed, but typically for unit vectors we would present them in decimal form to a reasonable number of significant figures.
Given a vector with a magnitude of 7.3 m and a direction of 250° counterclockwise from the positive x-direction, let's perform the calculations:
First, we convert the degree measure to radians because the trigonometric functions in most mathematical equations use radian measure:
[tex]\[ 250° = \left(\frac{250}{360}\right) \times 2\pi \approx 4.3633 \text{ radians} \][/tex]
(Here we're approximating the radian measure, typically you should keep it in terms of π or use a calculator to get an exact value).
The x and y components of a vector with an angle θ are found by:
[tex]\[ x = r \cos(\theta) \][/tex]
[tex]\[ y = r \sin(\theta) \][/tex]
Where:
- r is the magnitude of the vector,
- θ is the angle in radians measured counterclockwise from the positive x-axis.
Using the given magnitude r = 7.3 m and our converted angle θ = 4.3633 radians, we get the components:
[tex]\[ x = 7.3 \cos(4.3633) \][/tex]
[tex]\[ y = 7.3 \sin(4.3633) \][/tex]
Since the angle is more than 180° but less than 270°, we should expect both the x and y components to be negative (we are in the third quadrant of the coordinate system).
The x-component is A, which is the horizontal component, and here x would be:
[tex]\[ A = 7.3 \cos(4.3633) \approx -7.3 \times 0.342 \approx -2.4966 \][/tex]
(Here, I rounded the value of cos(250°), which is cos(4.3633 radians) to 0.342 for illustration purposes.)
The y-component is B, which is the vertical component, and here y would be:
[tex]\[ B = 7.3 \sin(4.3633) \approx -7.3 \times 0.9397 \approx -6.85981 \][/tex]
(Here, I rounded the value of sin(250°), which is sin(4.3633 radians) to 0.9397 for illustration purposes.)
To convert these components into unit vector notation, we will create a vector whose components are these values divided by the vector's magnitude. Since we're dealing with a unit vector, and its magnitude is 1, the components themselves represent the unit vector.
Unit vector notation:
[tex]\[ \vec{u} = \frac{1}{7.3}\vec{v} = \left(\frac{x}{7.3}, \frac{y}{7.3}\right) = \left(\frac{-2.4966}{7.3}, \frac{-6.85981}{7.3}\right) \][/tex]
Simplify the fractions to get the final unit vector notation:
[tex]\[ \vec{u} = (-0.342, -0.9397) \][/tex]
We could round these numbers as needed, but typically for unit vectors we would present them in decimal form to a reasonable number of significant figures.
