Vectors and Scalars (Grade 10 NSC Matric Physical Sciences): Revision Notes

Properties of Vectors

Introduction to vector properties

Vectors are special mathematical quantities that have both magnitude (size) and direction. Understanding their mathematical properties is essential for solving physics problems involving forces, displacement, velocity, and acceleration.

Equality of vectors

When comparing vectors, we need to check two important characteristics before we can say they are equal.

For example, if we have two forces, $F_1 = 20$ N in the upward direction and $F_2 = 20$ N in the upward direction, then we can say that $F_1 = F_2$. Both vectors must match exactly in both size and direction.

Negative vectors

Just like scalar quantities can be positive or negative, vectors can also have positive or negative values. The sign tells us about the direction relative to a chosen reference.

When solving problems, we choose a reference positive direction first. For example, if we define upward as positive, then a force of 30 N downwards would be written as $F_1 = -30$ N. The negative sign shows us that the vector points in the opposite direction to our chosen positive reference.

Addition and subtraction of vectors

Adding vectors

When we add vectors together, we must consider both their magnitudes and their directions. This makes vector addition different from simple arithmetic.

Let's understand this through a practical example. Imagine you and a friend are trying to move a heavy box. You stand behind it and push forwards with a force $F_1$, while your friend stands in front and pulls towards them with a force $F_2$. Since both forces act in the same direction (forwards), the total force is:

$F_{Total} = F_1 + F_2$

The concept becomes clearer when we think about displacement vectors. Displacement describes the change in an object's position - it points from the initial position to the final position.

When we add vectors graphically, we place the second vector at the end of the first vector. The resultant vector goes from the starting point (tail of the first vector) to the end point (head of the second vector).

Subtracting vectors

Vector subtraction occurs when forces or displacements act in opposite directions. Let's return to our box example, but this time imagine you're not communicating properly. You stand behind the box and pull it towards you with force $F_1$, while your friend stands at the front and pulls it towards them with force $F_2$. Now the forces act in opposite directions.

When vectors act in opposite directions, we define one direction as positive and treat the opposite direction as negative. The total force becomes:

$F_{Total} = F_2 + (-F_1) = F_2 - F_1$

The resultant vector

When we combine multiple vectors, we get a single vector that has the same overall effect.

We can understand this concept through two scenarios:

The resultant force in each case produces the same effect as the original individual forces combined.

Equilibrant

There's a special vector related to the resultant that's important in physics problems.

The equilibrant is useful because when you add the resultant vector and the equilibrant together, they cancel each other out (the answer is always zero). This concept is important in equilibrium problems where forces balance out.

For our previous examples:

• When $F_R = 35$ N to the right, the equilibrant $F_E = 35$ N to the left
• When $F_R = 5$ N to the left, the equilibrant $F_E = 5$ N to the right

Techniques of vector addition

There are two main approaches to adding vectors: graphical techniques and algebraic techniques. Understanding both methods helps you solve different types of problems effectively.

Graphical techniques

Graphical methods involve drawing accurate scale diagrams to represent individual vectors and find their resultants. The most common graphical method is the head-to-tail method.

This visual approach helps you understand how vectors combine and is particularly useful when working with vectors that aren't along the same line.