The vector v has a direction given by (-4, 27). Find the direction angle for v.a) 81.6 degrees b) 98.4 degrees c) 261.6 degrees d) 278.4 degrees

To find the direction angle of the vector v with the given direction (-4, 27), we need to use trigonometry. The direction angle is usually measured from the positive x-axis to the vector in a counter-clockwise direction.

Since the vector has a horizontal component of -4 and a vertical component of 27, we can use the arctangent function (also known as the inverse tangent or tan^(-1)) to find the angle θ. We must be careful, as the vector is not in the standard position (it is not starting at the origin and going into the first quadrant). Because the horizontal component is negative and the vertical component is positive, the vector lies in the second quadrant.

The formula to find the angle with the positive x-axis is: θ = arctan(vertical component / horizontal component)

θ = arctan(27 / -4) θ = arctan(-6.75)

Using a calculator, we find that: θ ≈ -81.6 degrees

This angle is negative because arctan functions assume the vector is in the fourth quadrant if the horizontal component is negative. But since we know that the vector is in the second quadrant, we must add 180 degrees to this angle to find the correct direction angle (because we need the angle in a counter-clockwise direction from the positive x-axis).

θ_corrected = θ + 180 degrees θ_corrected = -81.6 degrees + 180 degrees θ_corrected = 98.4 degrees

Therefore, the direction angle for vector v is 98.4 degrees.