Given ||v|| = 5 theta=30 degrees, write the vector v.
Well, I guess we can call this vector a "30-degree-trig-a-tron"! To write it, we first need to find the components of this vector. For an angle of 30 degrees, we can use some trigonometry!
The magnitude of the vector, which is given as ||v|| = 5, represents the hypotenuse of a right triangle. Let's call the adjacent side "x" and the opposite side "y". We can use the cosine and sine of 30 degrees to find the values of "x" and "y", respectively.
cos(30 degrees) = x/5
sin(30 degrees) = y/5
Solving these equations, we find that x = 5 * cos(30 degrees) = 5 * (√3/2) = 5√3/2, and y = 5 * sin(30 degrees) = 5 * (1/2) = 5/2.
So, the components of vector v are (5√3/2, 5/2).
Therefore, the vector v is written as v = 5√3/2 * i + 5/2 * j, where i and j are the unit vectors along the x and y-axes, respectively.
And that's the "lowdown on the trig-a-tron clown" answer for ya!
To write the vector v using ||v|| = 5 and theta = 30 degrees, we can use the following steps:
Step 1: Start with the general form of a vector in two-dimensional space, v = <x,y>.
Step 2: The magnitude of vector v is given as ||v|| = 5. This means that the length or magnitude of vector v is 5.
Step 3: Using the general magnitude formula for a two-dimensional vector, ||v|| = sqrt(x^2 + y^2), we can substitute the given magnitude value, 5, into the equation:
5 = sqrt(x^2 + y^2)
Step 4: To solve for x and y, we need to find the components of the vector. We can use the given angle theta = 30 degrees (counterclockwise from the positive x-axis) to find these components.
Step 5: In a right triangle, the x-component (horizontal projection) is given by |v| * cos(theta), and the y-component (vertical projection) is given by |v| * sin(theta).
Step 6: Substitute the values into the equations:
x = 5 * cos(30 degrees)
y = 5 * sin(30 degrees)
Step 7: Evaluate the trigonometric functions:
x = 5 * 0.866 = 4.33
y = 5 * 0.5 = 2.5
Step 8: Write the vector v using the components:
v = <4.33, 2.5>
To write the vector v given its magnitude ||v|| = 5 and the angle theta = 30 degrees, we can use the equations for the x and y components of a vector.
The x-component of v, denoted as v_x, can be found using the equation v_x = ||v|| * cos(theta), while the y-component, denoted as v_y, can be found using v_y = ||v|| * sin(theta).
Plugging in the given values, we have:
v_x = 5 * cos(30 degrees)
v_y = 5 * sin(30 degrees)
To find the vector v, we combine the x and y components as v = v_x*i + v_y*j, where i and j are the unit vectors in the x and y directions, respectively.
Using the values from above, we get:
v = (5 * cos(30 degrees))*i + (5 * sin(30 degrees))*j
Evaluating further:
v = (5 * 0.866)*i + (5 * 0.5)*j
v = 4.33*i + 2.5*j
So, the vector v is written as v = 4.33*i + 2.5*j.
u = 5 cos 30
v = 5 sin 30
so
V = 5 cos 30 i + 5 sin 30 J
of course cos 30 = sqrt 3 /2
and sin 30 = 1/2
so
V = (5 sqrt 3) /2 i + (5/2) j