Final answer:
The magnitude of the vector →u= ⌬2,−4⌫ is √20. The direction angle from the positive x-axis is approximately 296.57° or 5.18 radians, which is calculated by subtracting the arctan of the absolute y-value divided by the x-value from 360°.
Step-by-step explanation:
The question asks to find the magnitude and direction angle of the vector →u= ⌬2,−4⌫. To find the magnitude of a vector, we utilize the Pythagorean theorem:
Magnitude (|u|) = √(x² + y²) = √(2² + (-4)²) = √(4 + 16) = √20.
To find the direction angle θ from the positive x-axis, we calculate the arctangent of the y-component divided by the x-component of the vector (arctan(y/x)).
However, because the calculator usually gives the principal value of the arctangent, which in this case would be in the 4th quadrant, we need to adjust the angle since our vector falls in the 2nd quadrant. The angle in the 4th quadrant would be positive, but since the vector's angle is negative, we must subtract it from 360° or 2π radians to get the correct angle which points counterclockwise from the positive x-axis.
The correct direction angle θ is: θ = 360° - arctan(|-4|/2) = 360° - arctan(2) ≈ 360° - 63.43° = 296.57°.
The vector's magnitude is √20 and its direction angle is approximately 296.57° or in radians: θ ≈ 5.18 radians (since 360 degrees is equivalent to 2π radians).