4.1.2

# Vector Problems

Test yourself

## Resolving Vectors

You need to be able to resolve vectors into two components. That means separating the vector into two parts, each part at right angles to the other.

### Resolving into x and y components

• The vector shown here sits in the x and y plane.
• We can resolve this vector into two perpendicular components: vertical and horizontal.

### Resolving into x and y components

• The x (horizontal) component is given by:
• Fx = F cos θ
• The y (vertical) component is given by:
• Fy = F sin θ

## Resolving Along an Incline

For problems with an inclined plane (a slope), it can be useful to resolve forces along the direction of the slope and the direction perpendicular to the slope.

### Inclined plane

• A particle sits on a plane inclined at an angle θ.
• The weight of the cube is W and acts vertically downwards.
• There are no other forces acting.

### Resolve along plane

• The weight resolved along the plane is W sin θ, as shown by the triangle drawn.
• The direction of the force is shown by the direction of the arrow.

### Resolve perpendicular to the plane

• The weight resolved perpendicular to the plane is W cos θ, as shown by the triangle drawn.
• The direction of the force is shown by the direction of the arrow.

## Equilibrium

An object is in equilibrium if the resultant force acting on it is zero. An object in equilibrium may be stationary (still) or moving with constant velocity.

### Multiple forces

• The resultant force on an object in equilibrium is zero.
• This occurs when the forces form a closed triangle.

### Example

• The forces in the example form a closed triangle.
• Therefore the object is in equilibrium.

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