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# 2.1: Scalar and Vectors

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### 2.1: Scalar and Vectors

In mechanics, commonly used terms like force, speed, velocity, and work can be classified as either scalar or vector quantities. A scalar is a physical quantity that can be described by its magnitude alone and does not require any directional components. Examples of scalar quantities are mass, area, and length.

Scalar quantities with the same physical units can be added or subtracted according to the usual algebra rules for numbers. For example, a class ending 10 min earlier than 50 min lasts 50 min − 10 min = 40 min. Similarly, a 60 calorie serving of corn followed by a 200 calorie serving of donuts gives 60 calories + 200 calories = 260 calories of energy. When we multiply a scalar quantity by a number, we obtain the same scalar quantity but with a larger (or smaller) value. For example, if yesterday's breakfast had 200 calories of energy and today's breakfast has four times as much energy as yesterday, then today's breakfast has 4 × 200 calories = 800 calories of energy. Two scalar quantities can also be multiplied or divided by each other to form a derived scalar quantity. For example, if a train covers a distance of 120 km in 1 h, its speed is 120,000 m/3600 s = 33 m/s, where the speed is a derived scalar quantity obtained by dividing distance with time.

On the other hand, a vector quantity is a physical quantity that has both magnitude and direction. A vector can be graphically represented using an arrow. The arrow's length symbolizes the vector's magnitude, while the angle between it and a fixed axis determines its line of action. The head of the arrow represents the direction in which the vector is pointing. Examples of vector quantities include displacement, velocity, position, force, and torque.

Consider giving directions to a friend on a hiking trip for a rest point 6 km from the tent. The easiest way will be to identify which direction to reach the exact location; for example, 6 km southeast will be more helpful. This quantity is the displacement of the hiker.

Two vectors are equal if they have equal magnitudes and the same direction. Two vectors with identical directions are said to be parallel vectors. However, if a vector points in the opposite direction, exactly 180°, to the first vector, they are said to be antiparallel. Two vectors with directions perpendicular to each other are said to be orthogonal vectors.