You are provided with vectors O and P in the image below. Work out R = O + 3P, expressing the result in components.

Jack and Joy are business partners. Joy resides 80 km south and 60 m west of Jack's residence. Jack drives 42 km south, then 12 km east, and finally 22 km south, arriving at the meeting point. Joy uses a helicopter to arrive at the meeting point. Determine Joy's displacement vector. Express your answer using i) components where +x-axis points east and +y-axis points north. ii) using magnitude and direction.

The figure below shows three vectors and $\(\overline{O}\)$, $\(\overline{P}\)$ and $\(\overline{Q}\)$ in the xy plane with their lengths specified in arbitrary units. Find the sum of these three vectors and express the resultant in (i) its individual components, (ii) magnitude, and its angle relative to the positive y-axis.

A watch on a diver's wrist indicates that his diving time is 20s. His mechanical watch's second hand is 3 cm long. Using a coordinate system in which the y-axis is aligned with the 60s marker on the watch, what is the displacement vector of the second hand's tip between 0 and 20 seconds?

Given two vectors M = 6i - 4j, N = -5i + 2j, find the magnitude and direction of calculated vector P, where P = M + N.

For a vector, R = O + P + Q, what is the magnitude and direction of R given O = (4.2 km, 30° North of west), P = (2.5 km, west), and Q = (3.0 km, 45° North of east).

During a contest, three horses pull on a cart using cables in different directions. The aerial view of the forces on the cart is shown in the image. Cables exert forces using the tension vector, which is usually parallel to the rope. Tension-like forces are measured in newtons, N. A single horse using a single cable is used to replace the three horses. Determine the tension in the cable if it generates the same effects as the three cables. Give your answer using components with unit vectors.