A vector has an x component of -23.0 units and a y component of 27.0 units. Find the magnitude and direction of the vector.
? units at ? counterclockwise from the +x-axis
exactly where is arctan(-27/23). Hint, in quadrant II
To find the magnitude of the vector, we can use the Pythagorean theorem. The magnitude (or length) of a vector is given by:
|π| = β(ππ₯^2 + ππ¦^2),
where ππ₯ is the x-component and ππ¦ is the y-component of the vector.
Given that ππ₯ = -23.0 units and ππ¦ = 27.0 units, we can calculate the magnitude:
|π| = β((-23.0)^2 + (27.0)^2)
|π| = β(529 + 729)
|π| = β(1258)
|π| β 35.5 units
So, the magnitude of the vector is approximately 35.5 units.
To find the direction of the vector, we can use trigonometry. The direction of a vector is typically measured counterclockwise from the positive x-axis.
The angle (ΞΈ) can be calculated using the arctangent function:
ΞΈ = arctan(ππ¦ / ππ₯)
ΞΈ = arctan(27.0 / -23.0)
ΞΈ β -50.0 degrees
Since the y-component is positive and the x-component is negative, the vector is in the second quadrant. The direction is 50.0 degrees counterclockwise from the positive x-axis in the second quadrant.
Therefore, the magnitude of the vector is approximately 35.5 units, and the direction is 50.0 degrees counterclockwise from the positive x-axis.
To find the magnitude and direction of the vector, we can use the Pythagorean theorem and trigonometry.
1. Magnitude of the vector:
The magnitude (or length) of a vector is found by taking the square root of the sum of the squares of its components. In this case, we have an x-component of -23.0 units and a y-component of 27.0 units.
To calculate the magnitude (M) of the vector:
M = sqrt((-23.0)^2 + (27.0)^2)
Using a calculator:
M β 35.0 units
Therefore, the magnitude of the vector is approximately 35.0 units.
2. Direction of the vector:
The direction of a vector is often given in terms of an angle measured counterclockwise from a reference axis. In this case, we need to find the angle at which the vector points with respect to the positive x-axis.
To calculate the direction (ΞΈ) of the vector:
ΞΈ = arctan(y/x)
Substituting the given values:
ΞΈ = arctan(27.0 / -23.0)
Using a calculator:
ΞΈ β -48.37 degrees
Note: The negative sign indicates that the angle is measured counterclockwise from the negative x-axis.
Therefore, the magnitude of the vector is approximately 35.0 units, and it is at approximately -48.37 degrees counterclockwise from the +x-axis.