# PHY C1: Scalars & Vectors

Today we’ll be covering:

• Scalars & Vectors
• Resultants of Vectors
• Relative Vectors
• Resolution of Vectors

Let’s go!

What is a Scalar Quantity?
A physical quantity that has a magnitude, but not a direction.
Examples: distance, speed, current, work

What is a Vector Quantity?
A physical quantity that has both a magnitude & a direction.
Examples: displacement, velocity, momentum, electric field strength

Question: why is kinetic energy (½mv²) a scalar, while its component velocity (v) is a vector?
A: Because the velocity is SQUARED, thus if it were negative, – times – cancel out. The energy is always positive.

What are Resultant Vectors?
The combined effect of two or more different vectors (which could have different magnitudes & directions, or the same).
These vectors could be forces acting on an object, or the velocity of a moving object, etc.
If the vectors are acting on the same PLANE, they are known as COPLANAR VECTORS.

How do you calculate the resultant of multiple Coplanar Vectors?
Let’s say you have 2 vectors, Vector A & Vector B.
If both are acting in the same direction, then
Resultant = A + B

If they are acting in opposite directions, then
Resultant = A – B
OR
Resultant = B – A
Depending on which direction you refer to as “positive”.

How do you calculate the resultant of two vectors not on the same plane?
By drawing a VECTOR TRIANGLE (Tip-to-Tail Method).

Vector Triangles:
Let’s say you have 2 vectors, Vector A & Vector B, acting on a point. By rearranging the vectors & joining them from Tip to Tail, you can complete the triangle to find the resultant!

In A-Levels, we will have to utilise these theorems/rules to help find the resultant, depending on given information:
1. Pythagoras Theorem: a² + b² = c²
2. Cosine Rule: a² = b² + c² – 2bc cos A
3. Sine Rule: (sin A)/a = (sin B)/b

What are Relative Vectors?
Vectors which are expressed from the PERSPECTIVE (or FRAME) of another vector.
Examples of situations involving Relative Vectors:

1. An object in motion changing its direction of motion.
The CHANGE in Velocity is the initial velocity seen from the perspective of the new velocity.
2. Two objects in motion at different velocities/directions.
The RELATIVE Velocity of one object from the perspective of the other.

To calculate Relative Vectors, SUBTRACT the initial vector from the new vector (the current frame of reference).

What is the Resolution of Vectors?
Splitting up a Vector into its COMPONENTS.
Usually, these components are expressed as HORIZONTAL & VERTICAL components, which are 90 degrees from each other.

The Resultant of multiple vectors can also be calculated by resolving each vector into its components:
1. Visualising the vectors acting on the point/object
2. Resolve each vector into its horizontal & vertical components
3. Adding/subtracting all the coplanar components
4. Find the resultant vector of the remaining horizontal & vertical components