If v = (2, -5), find the magnitude and direction angle of 2v.
To find the magnitude of 2v, we need to multiply the components of v by 2 and then take the square root of the sum of the squares.
2v = 2(2, -5) = (4, -10)
Magnitude of 2v = √(4^2 + (-10)^2) = √(16 + 100) = √116 = 2√29
To find the direction angle of 2v, we use the formula:
Direction angle = tan^(-1)(y/x)
Here, x = 4 and y = -10
Direction angle = tan^(-1)(-10/4) = tan^(-1)(-5/2) ≈ -68.2 degrees (rounded to one decimal place).
Therefore, the magnitude of 2v is 2√29 and the direction angle is approximately -68.2 degrees.
To find the magnitude of 2v, we can multiply the magnitude of v by 2.
First, let's find the magnitude of v. The magnitude of a vector (x, y) can be found using the formula: magnitude = sqrt(x^2 + y^2).
So, the magnitude of v = sqrt((2^2) + (-5^2)) = sqrt(4 + 25) = sqrt(29).
Now, to find the magnitude of 2v, we multiply this magnitude by 2:
Magnitude of 2v = 2 * sqrt(29).
To find the direction angle of 2v, we can use the formula:
Direction angle = invtan(y / x).
For v = (2, -5), the direction angle = invtan((-5) / (2)) = invtan(-2.5).
So, the magnitude of 2v is 2 * sqrt(29), and the direction angle is invtan(-2.5).
Note: The direction angle is usually measured in radians or degrees, so you may need to convert it depending on the requirements of your question.