cFinal answer:
To find the magnitude and direction of u + v + w, add the x-components and the y-components separately to get (-19.969, 2.199). Then, use the Pythagorean theorem and inverse tangent to find the magnitude and direction: 20.130 and -6.429 degrees.
Step-by-step explanation:
To find the magnitude and direction of u + v + w, we need to add the vectors u, v, and w together. First, let's find the components of each vector using trigonometry.
For u, the x-component is 35*cos(40°) = 26.706 and the y-component is 35*sin(40°) = 22.337. Similarly, for v, the x-component is 13*cos(180°) = -13 and the y-component is 13*sin(180°) = 0. Lastly, for w, the x-component is 40*cos(310°) = -32.675 and the y-component is 40*sin(310°) = -20.138.
Now, we can add the x-components and the y-components separately. The x-component of u + v + w is 26.706 + (-13) + (-32.675) = -19.969 and the y-component is 22.337 + 0 + (-20.138) = 2.199.
Finally, we can find the magnitude and direction of the resultant vector using the Pythagorean theorem and inverse tangent. The magnitude is sqrt((-19.969)^2 + 2.199^2) = 20.130 rounded to the thousandths place. The direction is atan(2.199/(-19.969)) = -6.429 degrees rounded to the nearest degree.