Physics SCALARS AND VECTORS
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### Topics Covered

• Introduction
• Scalars
• Vectors
• Position Vector
• Displacement Vector
• Equality of Vectors

### Introduction

In order to describe motion of an object in two dimensions (a plane) or three dimensions (space), we need to use vectors to describe the physical quantities - position, displacement, velocity and acceleration etc.

### Scalars and Vectors

In physics, we can classify quantities as scalars or vectors. Basically, the difference is that a direction is associated with a vector but not with a scalar.

\color{fuchsia} \mathbf\ul"SCALAR QUANTITY"

\color{purple} "★ DEFINITION ALERT"

\color{purple} ✍️ A scalar quantity is a quantity with magnitude only. It is specified completely by a single number, along with the proper unit.

\color{green}"Examples" - Distance, Mass, Temperature, Volume etc.

\color{green} ★ \color{green} \mathbf(KEY \ CONCEPT)

\color{green} ✍️ Scalars can be added, subtracted, multiplied and divided just as the ordinary numbers. (Rules of ordinary algebra)

\color{green} ✍️ Addition and subtraction of scalars make sense only for quantities with same units. However, you can multiply and divide scalars of different units.

\color{fuchsia} \mathbf\ultext(VECTOR QUANTITY)

\color{purple} "★ DEFINITION ALERT"

\color{purple} ✍️ A vector quantity is a quantity that has both a magnitude and a direction and obeys the text(triangle law of addition) or equivalently the text(parallelogram law of addition). So, a vector is specified by giving its magnitude by a number and its direction.

\color{green}"Examples" - Displacement, Velocity, Acceleration, Force, etc.

\color{green} ★ \color{green} \mathbf(KEY \ CONCEPT)

\color{green} ✍️ To represent a vector, we use an arrow placed over a letter, say vecv.

\color{green} ✍️ The magnitude of a vector is often called its absolute value, indicated by |vecv| = v

### Position and Displacement Vectors

\color{fuchsia} \mathbf\ultext(POSITION VECTOR)

\color{purple} "★ DEFINITION ALERT"

\color{purple} ✍️ To describe the position of an object moving in a plane, we need to choose a convenient point, say O as origin.
Then the straight line having one end fixed to origin O and the other end attached to the position of object is called position vector.

\color{green} ★ \color{green} \mathbf(KEY \ CONCEPT)

\color{green} ✍️ An arrow is marked at the head of this line.

\color{green} ✍️ It is represented by a symbol vecr.

\color{green}"Example" - Let P and P′ be the positions of the object at time t and t′, respectively [Fig. a]. We join O and P by a straight line. Then, vec(OP) is the text(position vector) of the object at time t. An arrow is marked at the head of this line. It is represented by a symbol vecr, i.e. vec(OP) = vecr.

\color{fuchsia} \mathbf\ultext(DISPLACEMENT VECTOR)

\color{purple} "★ DEFINITION ALERT"

\color{purple} ✍️A displacement is a vector whose length is the shortest distance from the initial to the final position of a point.

\color{green}"Example" - Point P′ is represented by another position vector, vec(OP′) denoted by vec(r′). If the object moves from P to P′ , the vector vec(PP′) (with tail at P and tip at P′ ) is called the text(displacement vector) corresponding to motion from point P (at time t) to point P′ (at time t′).

\color{green} ★ \color{green} \mathbf(KEY \ CONCEPT)

\color{green} ✍️ It is important to note that displacement vector is the straight line joining the initial and final positions.

\color{green} ✍️ Displacement does not depend on the actual path undertaken by the object between the two positions.
\color{green}"Example" - In Fig. b, the displacement vector is the same vec(PQ) for different paths of journey, say PABCQ, PDQ, and PBEFQ.

\color{green} ✍️ Therefore, the magnitude of displacement is either less or equal to the path length of an object between two points.

### Equality of Vectors

\color{purple} "★ DEFINITION ALERT"

\color{purple} ✍️ Two vectors vecA and vecB are said to be equal if, and only if, they have the same magnitude and the same direction.

\color{green}"Examples" - Figure (a) shows two equal vectors vecA and vecB. We can easily check their equality. Shift vecB parallel to itself until its tail Q coincides with that of A, i.e. Q coincides with O. Then, since their tips S and P also coincide, the two vectors are said to be equal. In general, equality is indicated as vecA = vecB.

Note that in Fig. (b), vectors vec(A′) and vec(B′) have the same magnitude but they are not equal because they have different directions.