Vectors & Scalars

In SEE-level science, you measured quantities like speed, mass, and temperature. But have you noticed that some quantities also need a direction to be fully described? When you say "a bus is moving at 80 km/h towards Pokhara," you are describing a vector quantity. This lesson takes your understanding of physical quantities to the next level.

Scalars vs Vectors

Scalar quantities have only magnitude (size). Examples: mass (50 kg), temperature (25 C), time (10 s), speed (80 km/h), energy.

Vector quantities have both magnitude and direction. Examples: displacement (5 km east), velocity (80 km/h towards north), force (10 N downward), acceleration.

Vector Representation

A vector is represented by an arrow. The length of the arrow represents magnitude, and the arrowhead shows direction. We write vectors as A (bold) or with an arrow above: A.

Vector Addition

Vectors cannot be added like ordinary numbers. Two methods are commonly used:

Triangle Law: Place the tail of the second vector at the head of the first. The resultant is drawn from the tail of the first to the head of the second.

Parallelogram Law: Place both vectors at the same point. Complete the parallelogram. The diagonal from the common point gives the resultant.

Resultant magnitude: R = sqrt(A² + B² + 2AB cos theta), where theta is the angle between A and B.

Resolution of Vectors

Any vector can be split into two perpendicular components:

• Horizontal component: Ax = A cos theta
• Vertical component: Ay = A sin theta

Worked Example: A force of 20 N acts at 30 degrees to the horizontal. Find its components.

• Fx = 20 cos 30 = 20 x 0.866 = 17.32 N
• Fy = 20 sin 30 = 20 x 0.5 = 10 N

Unit Vectors

A unit vector has magnitude 1 and indicates direction only. The standard unit vectors are i (x-axis), j (y-axis), and k (z-axis). Any vector can be written as A = Ax i + Ay j + Az k.

Dot and Cross Product (Introduction)

Dot product (scalar product): A . B = AB cos theta. Result is a scalar. Example: Work = F . d.

Cross product (vector product): A x B = AB sin theta (direction given by right-hand rule). Result is a vector. Example: Torque = r x F.

Key Takeaways

• Scalars have magnitude only; vectors have magnitude and direction
• Vectors are added using triangle or parallelogram law
• Any vector can be resolved into perpendicular components
• Dot product gives a scalar; cross product gives a vector