Vector D's components are -7.0 cm (x) and 0 cm (y), while vector E's components are approximately 6.01 cm (x) and 6.01 cm (y). The resultant vector D+E has a magnitude of approximately 6.07 cm and a direction of 99.37 degrees from the positive x-axis.
To find the x and y components of vectors D and E, we use basic trigonometry. Vector D has a magnitude of 7.0 cm and points at 180 degrees, indicating a direction directly to the left on the x-axis. Therefore, its x-component is -7.0 cm (as it points left/backwards along the x-axis), and the y-component is 0 cm (as there is no displacement vertically).
For vector E, which has a magnitude of 8.5 cm and points at a 45-degree angle, we split this into x and y components using cosine and sine respectively:
Ex = 8.5 cm * cos(45°) = 6.01 cm (approximately)
Ey = 8.5 cm * sin(45°) = 6.01 cm (approximately)
To determine the magnitude and direction of D+E, we first add the x-components and y-components respectively:
Total x-component = Dx + Ex = -7.0 cm + 6.01 cm = -0.99 cm
Total y-component = Dy + Ey = 0 cm + 6.01 cm = 6.01 cm
The magnitude can be found using the Pythagorean theorem, and the angle using the inverse tangent function (tan-1) applied to the ratio of y-component over x-component.
Calculating the magnitude:
Magnitude = √((-0.99 cm)2 + (6.01 cm)2) = 6.07 cm (approximately)
Calculating the direction:
Direction angle = tan-1(6.01 cm / -0.99 cm) ≈ -80.63° (approximately)
Since the x-component is negative and the y-component is positive, this angle is measured in the second quadrant, meaning we must add 180° to get the angle from the positive x-axis, resulting in 99.37°.