Adding Vectors by Components Practice Problems

A physics instructor directs three students to pull a ring, as shown in the image. The students have different abilities, hence exerting different pulls. One student applies 4.0 N on the first cable, and a second student exerts 6.0 N on the second cable. Determine the pull a third student must apply on the third cable to keep the ring stationary. Express the direction as an angle measured from the negative x-axis.

A position vector lies in the second quadrant. The magnitude of the vector is 18 km. The x-component is - 10 km. Determine the y-component of the vector.

The acceleration vector of a light particle moving in the wind is directed at 65 degrees below the negative x-axis. The y-component of the acceleration is -8.8 m/s². Determine the x-component of the acceleration.  

An element in cement dust has a diameter of 10 μm. The element has a uniform speed of 2.5 cm/s. The motion of the element near a window is shown in the image below, from points O to S. Calculate the element's average velocity magnitude and direction for the entire motion. 

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.

A motorcycle race occurs on rugged ground. At 24 minutes after starting the race, a biker has a displacement R₁ = (600 m, west) + (500 m, South) + (10 m, vertical). 50 minutes from the start of the race, the biker has a displacement R₂ = (30000 m, west) + (25000 m, south) - (8 m, vertical). Determine the displacement magnitude of the biker at 50 minutes.

A building inspector surveying a group of multi-story buildings has a displacement R₁ = (1500 m, west) + (2000 m, South) + (100 m, vertical). In another instance, the inspector has a displacement R₂ = (1000 m, west) + (1500 m, south) - (200 m, vertical). If the inspection ends at the moment R₂ was taken, determine the height gained/lost by the inspector relative to the beginning point.

A circular driveway has a diameter of 10.0 m. A child circles the driveway at a constant speed. Use an x-y coordinate system whose origin is located at the center of the driveway; the +x-axis points to the east, and +y-axis points to the north. The child has an initial position (x,y) = (-3 m,4 m) and walks in a counterclockwise direction around the driveway. After 1.75 turns, determine the child's displacement vector. Express the result using magnitude and direction.

Aaron takes the following route from his house to the library. The image is not drawn to scale. For the route shown, determine the magnitude and direction of his resultant displacement by using the method of components.

For the vectors C and D in the image, find the magnitude and direction (measured as a counterclockwise angle from the positive x-axis) of the vector difference D - C. Subtracting one vector from another (e.g. D - C) can be thought of as adding the "reverse" of that second D + (- C).

For the vectors C and D in the image, find the magnitude and direction (measured as a counterclockwise angle from the positive x-axis) of the vector difference C - D. Subtracting one vector from another (e.g. C - D) can be thought of adding the "reverse" of that second C + (- D).

For the vectors C and D in the figure, find the magnitude and direction (measured as a counterclockwise angle from the positive x-axis) of the vector sum C + D. The order in which vectors are added does not matter because vector addition is commutative.

Given vector $\overrightarrow{P}$, which has a length of 7.9 cm and points eastward, and vector $\overrightarrow{Q}$ , which has a length of 3.6 cm and points westward. Find the magnitude and direction of vector $\overrightarrow{R}$ , where $\overrightarrow{R}$ equals the sum of vectors $\overrightarrow{P}$ and $\overrightarrow{Q}$ .

Find $\overline{P}-2\overline{Q}$, utilising the vectors shown in the diagram,

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?

You are given M = 5i - 3j, N = -2i + 4j. Determine the component form of vector P, for P = M + N.

Using the vectors M = 3i - 2j and N = -2i + 6j, give the result of O = M - 2N using magnitude and direction form.

A toy has four forces acting on it. The unit of force used is Newton, N. The sum of forces on the toy is ΣF = F_a + F_b + F_c + F_d = 6.0 j N. Determine the values of F_b and F_c, expressing the result using components.

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

Work out the vector sum O = 3P + Q using P = 3i - j and Q = -4i + 6j. Express vector O using magnitude and direction form.

Evaluate the vector operation  R = M - 5N for M = 5i - 3j and N = -2i + j, expressing the result in components.

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).

Sketch the vector P = -5.0i + 4.0j. Mark the magnitude and direction of the vector on the drawing.

Work out the vector magnitude and direction of s = (12i + 9j) m/s. Draw the vector and mark the magnitude and direction.

Calculate the magnitude and direction M = -3.0i + 3.0j. Draw the vector and show the direction clearly.

Use a diagram to express the vector P = -4.0i - 6.0j. Label the magnitude and direction clearly on the diagram.

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.

Represent the vectors, M = 2i - j, N = -2i + 4j, and  O = 2 M - N on an x-y plane.

Use the components method to find R = P + Q using the vectors in the image. Report the magnitude and orientation of R.

Represent A = 10i + 6j graphically. Indicate its direction and magnitude in the image.

Electric forces have their sources on charges. The basic law of electrostatics states that like charges repel, unlike charges attract. A  group of students charged spheres and assembled them as shown. The forces on sphere 1 are; A 12.0 N attractive force from sphere 2 that is directed outward from sphere 1 to 2. A 22.0 N repulsive force from sphere 3, directly outward from sphere 3 to sphere 1. The last force is a 17.0 N attractive force from sphere 4 directed from sphere 1 toward sphere 4. Treating forces like vectors, determine the magnitude and direction of the sum of forces, ΣF, on sphere 1.