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Vector Components

Definition of Component

Components are vectors which, when combined, form a resultant. Orthogonal components are components that are at 90 degrees to each other. They are normally parallel to the axes of a reference system.

Definition of Resolution

Taking apart a vector into components is called "vector resolution" or "resolving the vector."

Example

A man pulls on his dog with a force of 70.0 N directed at an angle of +30.0 degrees to the horizontal. Find the horizontal and vertical components (x and y components) of this force.

ANSWER TO EXAMPLE:

Vector components are often drawn in the horizontal and vertical orientations.

Assigning "x" to "horizontal," we have

Fx = (70.0)cos30º = 60.6 N

Fy = (70.0)sin30º = 35.5 N

Vector Component and Resolution Problems

(See bottom of page for answers.)

A vector can also be resolved into x and y components.

1. Draw the x and y components of this vector.

Vector components can be drawn along any set of axes.

2. Draw the x and y components of this vector.

Forces are often resolved into horizontal and vertical components.

3. A force applied to a lawnmower. Draw the horizontal and vertical components of the applied force.

Force components may be drawn parallel to or perpendicular to a slope.

4. Gravity is acting on a box at rest on a ramp. Draw the components of gravity that are parallel (Fx) and perpendicular (Fy) to the slope.

5. A man standing 3 m above the water throws a rope to a surfer in the water 5 m away from him. He pulls in the rope at a speed of 1 m/s. How fast is the surfer traveling when he is 4 m from shore?

6. A vector has a magnitude of 25 m and an x component of 12 m. What is the angle the vector makes with the x axis?

7. The shadow of a pine tree is an isosceles triangle. The angle at the top of the triangle is 30º, and the base measures 2 meters across. How tall is the shadow?

8. Two balls roll along a flat surface. One has the velocity components Vx = 1.0 m/s, Vy = 2.0 m/s the other has the velocity components Vx = 2.0 m/s, Vy = 3.0 m/s. If both balls start at the same point what is the angle between their paths?

9. A vector has a magnitude of 70.0 cm and a y component of -55.0 cm. What are the two possibilities for its x component?

10. A man on a scaffold is 36.0 ft above the ground. He looks at an angle of 14.5 degrees below the horizontal to see a boulder. What is the horizontal distance from the man to the boulder?

11. A seal dives at an angle of 19.0 degrees below the horizontal. If the seal continues in a straight line for 140m, a)How deep is it and b)How far has it traveled horizontally?

12. A jet travels at a speed of 245 m/s. The plane is gaining height at a rate of 40.6 m/s. Determine the magnitude of the horizontal component of the plane's velocity.

ANSWERS

For solutions to all the problems on this page click here.

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