Find the magnitude and direction angle for the vector v = 5 cos 144°i + 5 sin 144°j.
To find the magnitude of the vector v, we need to calculate the square root of the sum of the squares of the components:
|v| = sqrt((5 cos 144°)^2 + (5 sin 144°)^2)
= sqrt((5)^2(cos^2(144°) + sin^2(144°)))
= sqrt((5)^2(1))
= sqrt(25)
= 5
To find the direction angle of the vector v, we first need to convert the vector components to rectangular form:
v = 5 cos 144°i + 5 sin 144°j
= 5(-0.8090)i + 5(0.5878)j
= -4.045i + 2.939j
Now, we can find the direction angle using the arctangent function:
θ = arctan(2.939 / -4.045)
≈ -35.30°
Therefore, the magnitude of vector v is 5 and the direction angle is approximately -35.30°.