A vector has an x-component of
−25.5
units and a y-component of 56.0 units. Find the magnitude and direction of the vector.Direction
° (counterclockwise from the +x-axis)
Y/X = 56/-25.5 = -2.19608
Ar = -65.5o = Rerence angle.
A = -65.5 + 180 = 114.5o = Direction.
Mag. = Y/sinA = 56/sin114.5 = 61.5 Units
NOTE: If you draw the vector diagram showing 56 on the positive Y-axis and -25.5 on the negative X-axis, you can see that the Direction lies between 90
and 180o. Therefore, -65.5o cannot be the direction.
To find the magnitude and direction of a vector, you can use the Pythagorean theorem and trigonometry. Here's how you can do it:
Step 1: Find the magnitude using the Pythagorean theorem. The magnitude of a vector is given by the formula:
Magnitude = sqrt(x^2 + y^2)
In this case, the x-component is -25.5 and the y-component is 56.0. Plug these values into the formula to find the magnitude:
Magnitude = sqrt((-25.5)^2 + (56.0)^2)
Magnitude = sqrt(650.25 + 3136)
Magnitude = sqrt(4786.25)
Magnitude ≈ 69.17 units
So, the magnitude of the vector is approximately 69.17 units.
Step 2: Find the direction of the vector. To find the direction, we can use trigonometry and the inverse tangent function (tan^-1). The direction is the angle between the vector and the positive x-axis.
Direction = tan^-1(y/x)
In this case, the x-component is -25.5 and the y-component is 56.0. Plug these values into the formula to find the direction:
Direction = tan^-1(56.0 / -25.5)
Direction ≈ -64.47°
Note: The angle is negative because it is measured counterclockwise from the positive x-axis.
So, the direction of the vector is approximately -64.47° counterclockwise from the positive x-axis.