You are given a vector in the xy plane that has a magnitude of 94.0 units and a y component of -44.0 units.
A. What are the two possibilities for its component?
For this part I got x = 83.1 and -83.1
B. Assuming the x component is known to be positive, specify the vector V which, if you add it to the original one, would give a resultant vector that is 61.0 units long and points entirely in the -x direction.
This is where I'm stuck. I tried working it out like this:
x^2 + y^2 = r^2
x^2 + (44)^2 = (61)^2
x = 42.25
But it's wrong. Though, I'm sure that the y component for this second vector should be positive 44 to cancel out with the -44 from the first vector to get the third one.
Please, help?
You need to add a vector with y component of +44 (to cancel the -44) and an x component of -(83.1 +61) = -144.1.
The equation you are using is not the right one. The vector you add is not the vector you end up with. For the x component, you start with 83.1 and end up with -61.
To find the x-component of the second vector, we can use the formula for the magnitude of a resultant vector:
|V|^2 = (x1 + x2)^2 + (y1 + y2)^2
Where:
|V| is the magnitude of the resultant vector (61.0 units)
x1 and y1 are the x and y components of the original vector (x1 is known to be positive)
x2 and y2 are the x and y components of the second vector
Substituting the given values:
(61.0)^2 = (83.1 unit + x2)^2 + (-44.0 units + y2)^2
Since we want the resultant vector to point entirely in the -x direction, the y-component of the second vector (y2) should be zero. This means y2 = 0. Substituting this, the equation becomes:
(61.0)^2 = (83.1 unit + x2)^2 + (-44.0 units)^2
Expanding and simplifying:
3772.1 = (6923.61 + 166.2x2 + x2^2) + 1936
3772.1 = 6923.61 + 166.2x2 + x2^2 + 1936
3772.1 = 8859.61 + 166.2x2 + x2^2
Rearranging the equation:
x2^2 + 166.2x2 + (8859.61 - 3772.1) = 0
x2^2 + 166.2x2 + 5087.51 = 0
This is a quadratic equation, which can be solved using the quadratic formula:
x2 = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = 166.2, and c = 5087.51. Substituting these values and solving for x2:
x2 = (-166.2 ± √(166.2^2 - 4(1)(5087.51))) / 2(1)
x2 = (-166.2 ± √(27624.84 - 20350.04)) / 2
x2 = (-166.2 ± √7274.8) / 2
x2 = (-166.2 ± 85.31) / 2
There are two possibilities for x2:
x2 = (-166.2 + 85.31) / 2 = -40.45 (approximately)
or
x2 = (-166.2 - 85.31) / 2 = -125.755 (approximately)
So, the two possibilities for the x-component of the second vector are approximately -40.45 and -125.755.
To solve part B of the problem, we can start by assuming the x-component of the vector V is positive, as stated. Let's call the x-component of V as Vx and the y-component as Vy.
We know that the resultant vector of adding the original vector to vector V should have a magnitude of 61 units and should point entirely in the -x direction.
To find the x and y components of vector V, we can set up the following equations:
Vx + 83.1 = -61
Vy - 44 = 0
From the first equation, we can solve for Vx:
Vx = -61 - 83.1
Vx = -144.1
From the second equation, we can solve for Vy:
Vy = 44
Therefore, the vector V has components Vx = -144.1 and Vy = 44. The resultant of adding this vector to the original one will give a vector pointing entirely in the -x direction with a magnitude of 61 units.