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.