# Vectors

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## Force as a vector quantity

• All forces can be described as object A pulls/pushes object B.
• All forces can be represented in both size and direction buy an arrow on a diagram.

Examples of forces: gravitational force, frictional force, normal contact force

• Physical quantities that have direction as well as size are called vectors.

Examples:

Scalars - mass, speed, length, distance, energy

Vectors - force, velocity, acceleration, displacement, field strength

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## Weight as a vector acting at the centre of gravity

• Gravitational forces act at a distance with no contact being necessary. These forces are always drawn as if they act at the centre of gravity of the object.
• The gravitational force acting on an object due to the planet or moon whose surface it is on is known as its weight.
• The centre of gravity of the object is the point at which its weight can be considered to act.
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## More vectors

speed scalar how fast an object is travelling

velocity vector how fast an object is travelling and in which direction it is travelling

• For an object moving along a straight line, positive (+) and negative (-) are usually used to indicate movement in opposite directions.
• As long as it is moving, the distance travelled by an object is increasing, but the displacement can increase or decrease and, like velocity, can have both positive and negative values.
• Displacement is a vector quantity. It specifies both the distance and the direction of an object moved from a fixed point.

average speed = distance travelled / time taken v = d / t

average velocity = displacement / time taken v = s / t

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• To add together two scalar quantities the normal rules of arithmetic apply. eg. 2kg + 3kg = 5kg
• To add vector quantities, both the size and direction have to be taken into account.

The sum, or resultant of two vectors such as two forces acting on a single object is the single vector that could replace the two and have the same effect.

How to find the resultant vector by diagram:

• Draw an arrow that represents one of the vectors in both size and direction.
• Starting where this arrow finishes, draw an arrow that represents the second vector in size and direction.
• The sum, or resultant of the two vectors is represented (in both size and direction) by the single arrow drawn from the start of the first arrow to the finish of the second arrow.

The size of the resultant force can also be calculated by using Pythagoras' theorem.

You can also use trigonometry to find the size of the angle between the resultant and another force.

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## Checking that the resultant is zero

• You can draw a vector diagram (as explained on the previous card) using multiple forces by drawing an arrow for each vector, starting each new arrow where the previous one finished. The resultant of these vectors is represented by the last arrow which is drawn to close the gap between the end of the last vector and the start of the first vector.
• If another arrow does not need to be drawn, because all of the vectors drawn make their own closed shape, it means that the resultant force for all of the vectors is zero.

Stable structure have a resultant force of zero, as it shows that they will not move anywhere.

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## Splitting a vector in two

• A vector has an effect in any direction except the one which is at right angles to it.

Sometimes a vector has two indepenent effects which need to be isolated.

• Just as the combined effect of two vectors acting on a single object can be calculated, two separate effects of a single vector can be found by splitting the vector into two components. Provided that the directions of the two components are chosen to be at right angles, each one has no effect in the direction of the otehr so they are considered to act independently.
• The process of splitting a vector into two components is known as resolving or resolution of the vector.

How to find the components:

• The component of a vector a at an angle A to its own angle is acosA.
• The component of a vector a at an angle (90° - A) to its own angle is asinA.
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