If v = (2, -5), find the magnitude and direction angle of 2v.
To find the magnitude of 2v, we can simply multiply the magnitude of v by 2. The magnitude of v is given by:
|v| = sqrt((2)^2 + (-5)^2) = sqrt(29)
So, the magnitude of 2v is:
|2v| = 2 * |v| = 2 * sqrt(29)
To find the direction angle of 2v, we need to determine the angle that 2v makes with the positive x-axis. We can find this angle using trigonometry. The direction angle, θ, is given by:
θ = arctan(v_y / v_x)
where v_x and v_y are the x and y components of v, respectively.
In this case:
θ = arctan((-5) / 2) = arctan(-2.5)
Using a calculator, we find that:
θ ≈ -68.2 degrees
Therefore, the magnitude of 2v is 2 * sqrt(29), and the direction angle of 2v is approximately -68.2 degrees.
To find the magnitude and direction angle of 2v, we can follow these steps:
Step 1: Multiply each component of v by 2.
2v = ((2)(2), (2)(-5))
2v = (4, -10)
Step 2: Calculate the magnitude of 2v using the formula:
Magnitude = √(x^2 + y^2)
Magnitude = √(4^2 + (-10)^2)
Magnitude = √(16 + 100)
Magnitude = √116
Magnitude ≈ 10.77
Step 3: Calculate the direction angle of 2v using the formula:
Direction angle = tan^(-1)(y / x)
Direction angle = tan^(-1)(-10 / 4)
Direction angle ≈ -68.2 degrees
Therefore, the magnitude of 2v is approximately 10.77 and the direction angle is approximately -68.2 degrees.