You are given a vector in the xy plane that has a magnitude of 90.0 units and a y component of -61.0 units.
a)What are the two possibilities for its x component?
b) Assuming the x component is known to be positive, specify the magnitude of the vector which, if you add it to the original one, would give a resultant vector that is 80.0 units long and points entirely in the −x direction.
c)Specify the direction of the vector.
a) To find the two possibilities for the x component, we can use the Pythagorean theorem. The magnitude of a vector in the xy plane with components x and y is given by:
Magnitude = sqrt(x^2 + y^2)
Given that the magnitude of the vector is 90 units and the y component is -61 units, we can substitute these values into the equation:
90 = sqrt(x^2 + (-61)^2)
Squaring both sides of the equation, we get:
8100 = x^2 + 3721
Rearranging the equation, we have:
x^2 = 8100 - 3721
x^2 = 4379
Taking the square root of both sides, we get:
x = ±sqrt(4379)
So, the two possibilities for the x component are x = sqrt(4379) and x = -sqrt(4379).
b) To find the magnitude of the vector that, when added to the original vector, gives a resultant vector that is 80 units long and points in the -x direction, we can set up the following equation:
Original magnitude + Magnitude to be added = 80
Given that the original magnitude is 90 units, we can substitute this value into the equation:
90 + Magnitude to be added = 80
Subtracting 90 from both sides of the equation, we have:
Magnitude to be added = 80 - 90
Magnitude to be added = -10
So, the magnitude of the vector that, when added to the original vector, gives a resultant vector that is 80 units long and points in the -x direction is -10 units.
c) To specify the direction of the vector, we need to find the angle it makes with the positive x-axis. We can use trigonometry to find this angle. The angle can be found using the tangent function:
angle = arctan(y / x)
Given that the y component is -61 and the x component could be either sqrt(4379) or -sqrt(4379), we can substitute these values into the equation:
angle = arctan(-61 / sqrt(4379))
angle = arctan(-61 / -sqrt(4379))
Using a calculator or reference table, we can find the approximate value of the angle:
angle ≈ -52.48°
So, the direction of the vector is approximately -52.48° with respect to the positive x-axis.