In the previous section we saw position vector. In this section we will see displacement vector.
In the next section, we will see addition and subtraction of vectors.
Also, later in this section we will see equality of two vectors and multiplication of a vector by a scalar.
In the previous section we saw this:
• When the reading in the stop watch is t1, the object is at P1
• When the reading in the stop watch is t2, the object is at P2
We will proceed from here:
1. It is clear that, within a duration of (t2-t1) s, the object is displaced from P1 to P2
2. Join P1 and P2 using a straight line. This is shown in fig.4.3(a) below:
3. An arrow (pointing towards P2) should be marked on the line P1P2
■ Then $\vec{P_1 P_2}$ is called the displacement vector corresponding to the motion from point P1 (at time t1) to point P2 (at time t2)
4. The direction of the arrow on the line P1P2 in the fig.4.3(a) is important
Based on the direction of that arrow, we say these:
• Tail of the displacement vector $\vec{P_1 P_2}$ is at P1
• Tip of the displacement vector $\vec{P_1 P_2}$ is at P2
5. Consider fig.4.3(b) above
• $\vec{AB}$ is a displacement vector
• That is., initial position of an object is A and it's final position is B
6. There are several paths possible for the object to get from A to B. Some of them are:
ACB, ADB, AEFB etc.,
7. Which ever be the path, We can consider only the straight line joining A and B
■ The displacement vector is the straight line joining the initial and final positions
■ The displacement vector does not depend on the actual path taken by the object
■ So the magnitude of the displacement vector is always less than or equal to the actual path length followed by the object
8. Note the words 'less than or equal to' carefully
• If the object indeed takes the straight line path between A and B, displacement will be equal to path length
• If the path taken is any one other than that straight line, displacement will be surely less than the path length
• This is because:
♦ there is only one straight line possible between any two points and
♦ the distance through that straight line will be the shortest distance
2. To whatever position it is moved, two conditions must be satisfied:
(i) It's magnitude does not change
(ii) It's direction does not change
♦ 'No change in direction' implies that, they are all parallel
3. Vectors which can be moved around in this way are called free vectors
• Vectors which cannot be moved around in this way are called localized vectors
■ For localized vectors, the point of application cannot change
4. Let us see some examples:
(i) Consider rain falling at an angle
• The velocity of the rain drops can be represented by a vector. This vector will be at an angle
• This vector can be moved around because rain is falling on a large area
• Similarly, the velocity vector of a wind can also be moved around
• They are free vectors
(ii) Now consider a force vector which is applied to open a door
• We normally apply the force vector on the door handle
• Instead of the handle, if we apply the same force vector (with no change in magnitude and direction) at a point near the hinge, the effect will be different
• So this force vector cannot be moved around. It is a localized vector
• For our present discussion, we consider free vectors only
(i) The two vectors should have the same magnitude
(i) The two vectors should have the same direction
2. We can easily check whether two vectors are equal or not.
(i) Consider fig.4.4(a) below:
• Two vectors $\vec{AB}$ and $\vec{PQ}$ are given.
(ii) Shift $\vec{PQ}$ parallel to itself. This is shown in fig.4.4(b)
(iii) Shift it until the tail P coincides with the tail A of $\vec{AB}$.
(iv) In this situation, if the tip Q of $\vec{PQ}$ coincides with the tip B of $\vec{AB}$, then the two vectors are equal
• We can write: $\vec{AB}$ = $\vec{PQ}$.
3. (i) Consider fig.4.4(c) above.
• Two vectors $\vec{MN}$ and $\vec{UV}$ are given.
(ii) Shift $\vec{UV}$ parallel to itself. This is shown in fig.4.4(d)
(iii) Shift it until the tail U coincides with the tail M of $\vec{MN}$.
(iv) In the fig., we find that, even though the two vectors have the same magnitude, their tips will not coincide. The two vectors are not equal. This is because, their directions are not the same.
2. Let us denote such a number by $\lambda$ (the Greek letter lambda).
• What happens when we do the multiplication?
3. The answer can be written in two statements:
(i) The magnitude of $\vec{A}$ will be factored by $\lambda$
(ii) The direction of $\vec{A}$ will remain unchanged
4. Initially, the magnitude of $\vec{A}$ is $\left | \vec{A} \right |$
So after multiplication, the magnitude becomes λ$\left | \vec{A} \right |$
5. In short we get a new vector: λ$\vec{A}$
• Magnitude of the new vector is λ$\left | \vec{A} \right |$
• Direction of the new vector is same as that of $\vec{A}$
■ Let us see some examples:
1. In fig.4.5(a) below, $\vec{A}$ is multiplied by 2
The resultant vector '$2\vec{A}$' has the following properties:
• Magnitude is twice that of $\vec{A}$
• Direction is same as that of $\vec{A}$
2. In fig.4.5(b), $\vec{A}$ is multiplied by -1
The resultant vector '-$\vec{A}$' has the following properties:
• Magnitude is same as that of $\vec{A}$
• Direction is opposite to that of $\vec{A}$
3. In fig.4.5(c), $\vec{A}$ is multiplied by -1.5
The resultant vector '-1.5$\vec{A}$' has the following properties:
• Magnitude is 1.5 times that of $\vec{A}$
• Direction is opposite to that of $\vec{A}$
In the previous section we saw this:
• When the reading in the stop watch is t1, the object is at P1
• When the reading in the stop watch is t2, the object is at P2
We will proceed from here:
1. It is clear that, within a duration of (t2-t1) s, the object is displaced from P1 to P2
2. Join P1 and P2 using a straight line. This is shown in fig.4.3(a) below:
 |
| Fig.4.3 |
■ Then $\vec{P_1 P_2}$ is called the displacement vector corresponding to the motion from point P1 (at time t1) to point P2 (at time t2)
4. The direction of the arrow on the line P1P2 in the fig.4.3(a) is important
Based on the direction of that arrow, we say these:
• Tail of the displacement vector $\vec{P_1 P_2}$ is at P1
• Tip of the displacement vector $\vec{P_1 P_2}$ is at P2
5. Consider fig.4.3(b) above
• $\vec{AB}$ is a displacement vector
• That is., initial position of an object is A and it's final position is B
6. There are several paths possible for the object to get from A to B. Some of them are:
ACB, ADB, AEFB etc.,
7. Which ever be the path, We can consider only the straight line joining A and B
■ The displacement vector is the straight line joining the initial and final positions
■ The displacement vector does not depend on the actual path taken by the object
■ So the magnitude of the displacement vector is always less than or equal to the actual path length followed by the object
8. Note the words 'less than or equal to' carefully
• If the object indeed takes the straight line path between A and B, displacement will be equal to path length
• If the path taken is any one other than that straight line, displacement will be surely less than the path length
• This is because:
♦ there is only one straight line possible between any two points and
♦ the distance through that straight line will be the shortest distance
Free vectors and Localized vectors
1. Consider $\vec{P_1 Q_1}$ shown in fig.4.3(c) above. It is moved around to different positions: P1Q1, P2Q2, P3Q3, etc.,2. To whatever position it is moved, two conditions must be satisfied:
(i) It's magnitude does not change
(ii) It's direction does not change
♦ 'No change in direction' implies that, they are all parallel
3. Vectors which can be moved around in this way are called free vectors
• Vectors which cannot be moved around in this way are called localized vectors
■ For localized vectors, the point of application cannot change
4. Let us see some examples:
(i) Consider rain falling at an angle
• The velocity of the rain drops can be represented by a vector. This vector will be at an angle
• This vector can be moved around because rain is falling on a large area
• Similarly, the velocity vector of a wind can also be moved around
• They are free vectors
(ii) Now consider a force vector which is applied to open a door
• We normally apply the force vector on the door handle
• Instead of the handle, if we apply the same force vector (with no change in magnitude and direction) at a point near the hinge, the effect will be different
• So this force vector cannot be moved around. It is a localized vector
• For our present discussion, we consider free vectors only
Equality of vectors
1. Two vectors are said to be equal only when both the conditions given below are satisfied:(i) The two vectors should have the same magnitude
(i) The two vectors should have the same direction
2. We can easily check whether two vectors are equal or not.
(i) Consider fig.4.4(a) below:
 |
| Fig.4.4 |
(ii) Shift $\vec{PQ}$ parallel to itself. This is shown in fig.4.4(b)
(iii) Shift it until the tail P coincides with the tail A of $\vec{AB}$.
(iv) In this situation, if the tip Q of $\vec{PQ}$ coincides with the tip B of $\vec{AB}$, then the two vectors are equal
• We can write: $\vec{AB}$ = $\vec{PQ}$.
3. (i) Consider fig.4.4(c) above.
• Two vectors $\vec{MN}$ and $\vec{UV}$ are given.
(ii) Shift $\vec{UV}$ parallel to itself. This is shown in fig.4.4(d)
(iii) Shift it until the tail U coincides with the tail M of $\vec{MN}$.
(iv) In the fig., we find that, even though the two vectors have the same magnitude, their tips will not coincide. The two vectors are not equal. This is because, their directions are not the same.
Multiplication of vectors by real numbers
1. Consider any vector $\vec{A}$. We can multiply it with any real number like 1.5, 2, -5, 10, 1500, 3.14, 1.732 etc.,2. Let us denote such a number by $\lambda$ (the Greek letter lambda).
• What happens when we do the multiplication?
3. The answer can be written in two statements:
(i) The magnitude of $\vec{A}$ will be factored by $\lambda$
(ii) The direction of $\vec{A}$ will remain unchanged
4. Initially, the magnitude of $\vec{A}$ is $\left | \vec{A} \right |$
So after multiplication, the magnitude becomes λ$\left | \vec{A} \right |$
5. In short we get a new vector: λ$\vec{A}$
• Magnitude of the new vector is λ$\left | \vec{A} \right |$
• Direction of the new vector is same as that of $\vec{A}$
■ Let us see some examples:
1. In fig.4.5(a) below, $\vec{A}$ is multiplied by 2
 |
| Fig.4.5 |
• Magnitude is twice that of $\vec{A}$
• Direction is same as that of $\vec{A}$
2. In fig.4.5(b), $\vec{A}$ is multiplied by -1
The resultant vector '-$\vec{A}$' has the following properties:
• Magnitude is same as that of $\vec{A}$
• Direction is opposite to that of $\vec{A}$
3. In fig.4.5(c), $\vec{A}$ is multiplied by -1.5
The resultant vector '-1.5$\vec{A}$' has the following properties:
• Magnitude is 1.5 times that of $\vec{A}$
• Direction is opposite to that of $\vec{A}$
In the next section, we will see addition and subtraction of vectors.
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