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Topics Covered

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


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.