what is the magnitude and direction of the vector U= -2i - 3j
To find the magnitude and direction of a vector, you can use some basic vector operations. In this case, the vector U is given as -2i - 3j.
Magnitude of a vector:
The magnitude of a vector represents its length or size. To find the magnitude of U, you can use the formula:
|U| = √(Ux^2 + Uy^2)
Where Ux is the component of U along the x-axis and Uy is the component of U along the y-axis.
In this case, Ux = -2 and Uy = -3:
|U| = √((-2)^2 + (-3)^2)
|U| = √(4 + 9)
|U| = √13
|U| ≈ 3.61
Therefore, the magnitude of vector U is approximately 3.61.
Direction of a vector:
The direction of a vector can be represented in various ways, such as using angles or unit vectors. One common way is to find the angle between the vector and the positive x-axis.
To find the angle θ, you can use the formula:
θ = arctan(Uy / Ux)
In this case, Ux = -2 and Uy = -3:
θ = arctan((-3) / (-2))
θ = arctan(1.5)
θ ≈ 56.31° (rounded to two decimal places)
Since the vector U has negative components along both the x and y-axis, the angle is in the second quadrant. Therefore, the direction of the vector U is approximately 56.31° (measured counterclockwise from the positive x-axis).
In summary, the magnitude of vector U is approximately 3.61, and its direction is approximately 56.31° counterclockwise from the positive x-axis.
sqrt (4 + 9) = sqrt 13
tan T = -2/-3 = 33.7 deg in quad 3
so
33.7 + 180 = 213.7 degrees