Final answer:
To find the angles made by the vector v = 40i - 30j with the positive x- and y-axes, we use trigonometry. The angle with the x-axis is approximately -36.87°, and the angle with the y-axis is approximately -53.13°. The unit vector in the direction of v can be found by dividing v by its magnitude.
Step-by-step explanation:
To determine the angles made by vector v = 40i - 30j with the positive x-axis and y-axis, we need to find the direction of the vector. The direction of a vector is given by the angle it makes with the positive x-axis or y-axis. We can find this angle using trigonometry.
The angle with the positive x-axis is given by tan⁻¹(vy/vx), and the angle with the positive y-axis is given by tan⁻¹(vx/vy).
For vector v = 40i - 30j, vx = 40 and vy = -30.
Angle with x-axis = tan⁻¹((-30)/40) = tan⁻¹(-3/4) ≈ -36.87°
Angle with y-axis = tan⁻¹((40)/(-30)) = tan⁻¹(-4/3) ≈ -53.13°
The unit vector n in the direction of v can be found by dividing v by its magnitude. The magnitude of v is given by |v| = √(vx² + vy²).
Plugging in the values, |v| = √((40)² + (-30)²) ≈ 50.
Therefore, the unit vector n in the direction of v is n = (v/|v|).