# Scalars and Vectors:-

Different physical quantities can be classified into the following two categories:

## (d) A vector is written with an arrowhead over its symbol like “”.Unit Vectors:-

A unit vector is a vector having a magnitude of unity. Its only purpose is to describe a direction in space. On x-y co-ordinate system   denote unit vector in positive x direction and  denotes unit vector in positive y direction.

## Equal Vectors:-Two or more, vectors are equal if they have the same magnitude (length) and direction, whatever their initial points. In the figure above, the vectors A and B are equal.Negative Vectors:-Two vectors which have same magnitude (length) but their direction is opposite to each, other called the negative vectors of each other. In figure above vectors A and C or B and C are negative vectors. Null Vectors:-A vector having zero magnitude an arbitrary direction is called zero vector or ‘null vector’and is written as = O vector. The initial point and the end point of such a vector coincide so that its direction is indeterminate. The concept of null vector is hypothetical but we introduce it only to explain some mathematical results.

### Properties of a null vector:-

(a) It has zero magnitude.

(b) It has arbitrary direction

(c) It is represented by a point.

(d) When a null vector is added or subtracted from a given vector the resultant vector is same as the given vector.

(e) Dot product of a null vector with any vector is always zero.

(f) Cross product of a null vector with any vector is also a null vector.

## Non-localized Vectors:- Vector whose initial point (tail) is  not fixed is said to be a non-localized or a free vector.

### Question 1:-

A ship sets out to sail a point 124 km due north. An unexpected storm blows the ship to a point 72.6 km to the north and 31.4 km to the east of its starting point. How far, and in what direction, must it now sail to reach its original destination?

### Assumption:-

We assume that the ship was initially at point O from where it was drawn to the new position A and finally it went to its destination to reach point D(shown in the figure below).

We assume that the position vector of point D relative to point A is given by vector such that the vertical and horizontal components of the vector describe the motion of the ship lately.

We also assume that the position vector of point D relative to point O is given by vector whereas the position vector of point  A relative to point O is given by vector .

For vector (refer diagram above, the red color vector),

Since this is a position vector for point D relative to O, it has only vertical vector component, given as:



We have shown the vertical component of vector in the figure above in red color to avoid confusion.

The horizontal vector component of this vector is zero and can be written as:



For vector  (refer diagram above, the blue color vectors),

Since the ship was driven 31.4 km east and 72.6 km north, the horizontal and vertical vector component of vector  is given as:





For vector  (refer diagram above, the black color vectors),

Since the ship was meant to go only in the north, the horizontal displacement of the ship, when it is at point A, from its destination is 31.4 east. Therefore the horizontal vector component of vector  can be written as:



The negative sign shows that the vertical vector components points in the west that is opposite to the direction of the unit vector .

The vertical vector component of vector  can be calculated from the difference between vertical vector component of vector  and the vertical vector component of vector  as:





Now, we have the components of vector , we can calculate the direction in which the ship must move and by what distance.

Let us assume that the angle subtended by vector  on negative x  axis is ϕ. The sign of the angle will indicate if the angle was measured clockwise or counterclockwise.

The angle can be calculated as:

ϕ = tan-1 (ay/ax)

Also from the components of vector  , we have that

ay = 51.4 km

ax = -31.4 km

Substituting the values in the equation above, we have

ϕ = tan-1 (51.4 km/-31.4 km)

= - 58.5˚

The negative sign indicates that the angle is measured in the counterclockwise direction.

Therefore the ship must go in the direction 58.5˚ north of west to reach its destination.

The distance that the ship needs to travel, to reach its destination is equal to the magnitude of vector . We can write the vector in the unit vector notation as:





The magnitude of this vector is:

|a| = √(ax)2 +(ay)2

= √(-31.4 km)2 +(51.4 km)2

= 60.2 km

Therefore the ship has to go 60.2 km in the direction of vector .

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