# Find the components of the vertical force Bold Upper FFequals=left angle 0 comma negative 4 right angle0,−4 in the directions parallel to and normal to the plane that makes an angle of StartFraction pi Over 3 EndFraction π 3 with the positive​ x-axis. Show that the total force is the sum of the two component forces.

Step-by-step explanation:

- A plane is oriented in a Cartesian coordinate system such that it makes an angle of ( π / 3 ) with the positive x - axis.

- A force ( F ) is directed along the y-axis as a vector < 0 , - 4 >

- We are to determine the the components of force ( F ) parallel and normal to the defined plane.

- We will denote two unit vectors: ( ) parallel to plane and ( ) orthogonal to the defined plane. We will define the two unit vectors in ( x - y ) plane as follows:

- The unit vector ( ) parallel to the defined plane makes an angle of ( 30° ) with the positive y-axis and an angle of ( π / 3 = 60° ) with the x-axis. We will find the projection of the vector onto the x and y axes as follows:

= < cos ( 60° ) , cos ( 30° ) >

- Similarly, the unit vector ( ) orthogonal to plane makes an angle of ( π / 3 ) with the positive x - axis and angle of ( π / 6 ) with the y-axis in negative direction. We will find the projection of the vector onto the x and y axes as follows:

- To find the projection of force ( F ) along and normal to the plane we will apply the dot product formulation:

- The Force vector parallel to the plane ( ) would be:

- Similarly, to find the projection of force ( ) normal to the plane we again employ the dot product formulation with normal unit vector (    ) as follows:

- To prove that the projected forces ( ) and ( ) are correct we will apply the vector summation of the two orthogonal vector which must equal to the original vector < 0 , - 4 >

.. proven

## Related Questions

The residents of a certain dormitory have collected the following data: People who live in the dorm can be classified as either involved in a relationship or uninvolved. Among involved people, 10 percent experience a breakup of their relationship every month. Among uninvolved people, 15 percent will enter into a relationship every month. What is the steady-state fraction of residents who are uninvolved

The steady state proportion for the U (uninvolved) fraction is 0.4.

Step-by-step explanation:

This can be modeled as a Markov chain, with two states:

U: uninvolved

M: matched

The transitions probability matrix is:

The steady state is that satisfies this product of matrixs:

being π the matrix of steady-state proportions and P the transition matrix.

If we multiply, we have:

Now we have to solve this equations

We choose one of the equations and solve:

Then, the steady state proportion for the U (uninvolved) fraction is 0.4.

Determine the maximized area of a rectangle that has a perimeter equal to 56m by creating and solving a quadratic equation. What is the length and width?

Area of rectangle =

Length of rectangle = 14 m

Width of rectangle = 14 m

Step-by-step explanation:

Given:

Perimeter of rectangle is 56 m

To find: the maximized area of a rectangle and the length and width

Solution:

A function has a point of maxima at if

Let x, y denotes length and width of the rectangle.

Perimeter of rectangle = 2( length + width )

Also, perimeter of rectangle is equal to 56 m.

So,

Let A denotes area of rectangle.

A = length × width

Differentiate with respect to x

Put

Also,

At x = 14,

So, x = 14 is a point of maxima

So,

Area of rectangle:

Length of rectangle = 14 m

Width of rectangle = 14 m

156.95

Step-by-step explanation:

length x width x height

Find the value of Z.

A. 3
B. 6\sqrt{2}
C. 2
D. 2\sqrt{2}

So you would first find the length of y.

To do that, use the Pythagorean's Theorem:

a²+b²=c²

in this case, c = 3, and b = 1.

a² + 1² = 3²
a² + 1 = 9
a² = 8
a = √8

Now to find z, use the Pythagorean's Theorem again:

a² + b² = c²

where √8 = a and 8 = b, and z = c.

√8² + 8² = z²
8 + 64 = z²
72 = z²
z = √72

To simplify this, take out the largest perfect square, or 36:

z = √36√2
z = 6√2

The range of the function f(x) = x + 5 is {7, 9}. What is the function’s domain?

the range is the y valuewe can rewrite the given function f(x)=x+5 to y=x+5and our given range are y=7 and y=9so sub in ur y values into the equation to find the x value which are the domain...y=x+5

The circumference of a circle is 28 in.What is the diameter of the circle?

Responses

28 over pi, in.

14 over pi, in.

square root of 28 over pi end root, in.

14π−−√ in.

I think it is 28/pi but I would like to make sure

The diameter οf the circle is 28/π inches οr apprοximately 8.89 inches (rοunded tο twο decimal places).

The fοrmula fοr the circumference (C) οf a circle is given by:

C = 2πr

where r is the radius οf the circle.

If the circumference οf the circle is 28 inches, we can sοlve fοr the radius by dividing bοth sides οf the equatiοn by 2π:

C/2π = r

Substituting the given value οf C = 28, we get:

r = 28/2π

r = 14/π

Finally, tο find the diameter (d) οf the circle, we multiply the radius by 2:

d = 2r

Substituting the value οf r = 14/π, we get:

d = 2(14/π) = 28/π

Therefοre, the diameter οf the circle is 28/π inches οr apprοximately 8.89 inches (rοunded tο twο decimal places).