In the previous section, we completed a discussion on differential equations. In this chapter, we will see vector algebra.
Some basics about vectors can be written in 2 steps:
1. Consider the length of a room. We can write:
Length of the room is 3.2 m
• Length involves only one value, which is the magnitude.
• Quantities which involve only the magnitude, are called scalars.
• A scalar quantity can be represented by a real number and a proper unit.
• Some examples are:
♦ A mass of 32 kg
♦ An area of 90 sq.cm
♦ A speed of 7 m/s
2. A quantities having both magnitude and direction are called vectors.
• Some examples are:
♦ A car moving with a velocity. We can write: velocity is 35 km/h towards east. Here, velocity is a vector quantity.
♦ A force applied on a body. We can write: Force is 25 N along a line which makes 30 degrees with the x-axis. Here, force is a vector quantity.
Let us see vectors in some more detail. It can be written in 8 steps:
1. Fig.26.1(a) below, shows a line $\small{l}$. It extents to infinity in both ways.
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| Fig.26.1 |
2. In fig.b, there is an arrowhead attached to the same line $\small{l}$. This arrowhead gives a sense of direction to the observer.
3. In fig.c the arrowhead is attached in the opposite direction. This arrowhead gives the sense of opposite direction to the observer.
4. In fig.d, two points A and B are marked on the same line $\small{l}$. AB is a line segment.
• Any line segment will have a definite length. That means, any line segment will have a magnitude.
• An arrowhead is also attached to the segment AB. This arrowhead gives the sense of direction.
• So the line segment AB in fig.d has both magnitude and direction. It is a vector.
5. The vector in fig.d can be denoted as: $\small{\vec{AB}}$ or as $\small{\vec{a}}$.
♦ $\small{\vec{AB}}$ is read as “vector AB”
♦ $\small{\vec{a}}$ is read as “vector a”
6. $\small{\vec{AB}}$ starts at A. So the point A is called the initial point of $\small{\vec{AB}}$
7. $\small{\vec{AB}}$ ends at B. So the point B is called the terminal point of $\small{\vec{AB}}$.
8. The distance between points A and B is called the magnitude of $\small{\vec{AB}}$. It is denoted as $\small{\left|{\vec{AB}}\right|}$
Note that, a length cannot be −ve. So we never write: $\small{\left|{\vec{AB}}\right| < 0}$
Position Vector
This can be explained in 4 steps:
1. In class 11, we have seen the three dimensional right handed rectangular coordinate system. Details here.
In that coordinate system, consider a point P. This is shown in fig.26.2 below:
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| Fig.26.2 |
♦ The coordinates of origin O are (0,0,0)
♦ The coordinates of P are (x,y,z)
2. In such a situation, we can draw $\small{\vec{OP}}$
• For $\small{\vec{OP}}$, O is the initial point and P is the terminal point.
• This $\small{\vec{OP}}$ is called the position vector of point P with respect to O.
3. Based on the discussions that we had in class 11, we are able to find the distance OP. Details here.
We get: $\small{OP = \sqrt{x^2 + y^2 + z^2}}$.
• So we can write: $\small{\left|{\vec{OP}}\right| = \sqrt{x^2 + y^2 + z^2}}$
4. Position vector is an important concept in science and engineering. For points A, B, C, . . . , we denote the position vectors as:
$\small{\vec{a},~\vec{b},~\vec{c},~.~.~.~,}$.
This is shown in fig.26.3 below:
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| Fig.26.3 |
Direction Cosines
There are three direction cosines. First we will see the direction cosine related to the x-axis. It can be written in 4 steps:
1. Fig.26.4 below shows a point P(x,y,z).
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| Fig.26.4 |
• The position vector $\small{\vec{OP}}$ makes an angle $\small{\alpha}$ with the +ve direction of the x-axis.
• For better clarity, the observer is looking at the coordinate system in the direction marked as I in fig.26.5 below.
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| Fig.26.5 Top view of rectangular coordinate system |
2. A perpendicular is dropped from P onto the xy-plane. Foot of this perpendicular is marked as P1.
• From P1, a perpendicular is dropped onto the x-axis. Foot of this perpendicular is marked as A.
3. Consider the plane containing the three point P, P1 and A.
• This plane will be perpendicular to the x-axis because, AP1 is perpendicular to the x-axis.
• Since this plane is perpendicular to the x-axis, any line in the plane will be perpendicular to the x-axis.
• So AP is perpendicular to the x-axis.
• Therefore, $\small{\angle{OAP} = 90^\circ}$
4. So triangle OAP is a right triangle with OP as hypotenuse. We get:
$\small{\begin{array}{ll} {~\color{magenta} 1 } &{{}} &{\cos \alpha} & {~=~} &{\frac{OA}{\left|\vec{OP} \right|} = \frac{x}{\left|\vec{OP} \right|}}
\\ {~\color{magenta} 2 } &{\Rightarrow} &{x} & {~=~} &{\cos \alpha\,\left|\vec{OP} \right|}
\\ \end{array}}$
Next we will see the direction cosine related to the y-axis. It can be written in 4 steps:
1. Fig.26.6 below shows the same point P(x,y,z).
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| Fig.26.6 |
• The position vector $\small{\vec{OP}}$ makes an angle $\small{\beta}$ with the +ve direction of the y-axis.
• For better clarity, the observer is looking at the coordinate system in the direction marked as II in fig.26.5 above.
2. We have the same point P1 as before.
• This time, the perpendicular from P1, is dropped onto the y-axis. Foot of this perpendicular is marked as B.
3. Consider the plane containing the three point P, P1 and B.
• This plane will be perpendicular to the y-axis because, BP1 is perpendicular to the y-axis.
• Since this plane is perpendicular to the y-axis, any line in the plane will be perpendicular to the y-axis.
• So BP is perpendicular to the y-axis.
• Therefore, $\small{\angle{OBP} = 90^\circ}$
4. So triangle OBP is a right triangle with OP as hypotenuse. We get:
$\small{\begin{array}{ll}
{~\color{magenta} 1 } &{{}} &{\cos \beta} &
{~=~} &{\frac{OB}{\left|\vec{OP} \right|} =
\frac{y}{\left|\vec{OP} \right|}}
\\ {~\color{magenta} 2 } &{\Rightarrow} &{y} & {~=~} &{\cos \beta\,\left|\vec{OP} \right|}
\\ \end{array}}$
Finally we will see the direction cosine related to the z-axis. It can be written in 3 steps:
1. Fig.26.7 below shows the same point P(x,y,z).
 |
| Fig.26.7 |
• The position vector $\small{\vec{OP}}$ makes an angle $\small{\gamma}$ with the +ve direction of the z-axis.
• For better clarity, the observer is looking at the coordinate system in the direction marked as I in fig.26.5 above.
2. This time, we do not need the point P1.
• This time, we directly drop the perpendicular from P onto the z-axis. Foot of this perpendicular is marked as C.
3. Since PC is perpendicular to the z-axis, triangle OPC is a right triangle with OP as hypotenuse. We get:
$\small{\begin{array}{ll}
{~\color{magenta} 1 } &{{}} &{\cos \gamma} &
{~=~} &{\frac{OC}{\left|\vec{OP} \right|} =
\frac{z}{\left|\vec{OP} \right|}}
\\ {~\color{magenta} 2 } &{\Rightarrow} &{z} & {~=~} &{\cos \gamma\,\left|\vec{OP} \right|}
\\ \end{array}}$
Now we can write the definitions. It can be written in 7 steps:
1. The angles $\small{\alpha, \beta~\text{and}~\gamma}$ that we saw above, are angles made with the +ve direction of the x, y and z axes respectively.
• They are called direction angles of the position vector $\small{\vec{OP}}$
2. We can take cosines of the above angles. We get:
$\small{\cos\alpha, \cos\beta~\text{and}~\cos\gamma}$
• These are called direction cosines of the position vector $\small{\vec{OP}}$
3. Usually we use short forms for denoting the direction cosines.
♦ $\small{\cos\alpha}$ is denoted by $\small{l}$
♦ $\small{\cos\beta}$ is denoted by $\small{m}$
♦ $\small{\cos\gamma}$ is denoted by $\small{n}$
4. There are short forms for the position vector also:
♦ $\small{\vec{OP}}$ is denoted by $\small{\vec{r}}$
♦ $\small{\left|\vec{OP} \right|}$ is denoted by $\small{r}$
5. Based on the above short forms, we can write:
$\small{\begin{array}{ll} {~\color{magenta} 1 } &{{}} &{x} & {~=~} &{\cos \alpha\,\left|\vec{OP} \right|}
\\ {~\color{magenta} 2 } &{\Rightarrow} &{x} & {~=~} &{lr}
\\ {~\color{magenta} 3 } &{} &{y} & {~=~} &{\cos \beta\,\left|\vec{OP} \right|}
\\ {~\color{magenta} 4 } &{\Rightarrow} &{y} & {~=~} &{mr}
\\ {~\color{magenta} 5 } &{} &{z} & {~=~} &{\cos \gamma\,\left|\vec{OP} \right|}
\\ {~\color{magenta} 6 } &{\Rightarrow} &{z} & {~=~} &{nr}
\\ \end{array}}$
6. So the coordinates (x,y,z) of the point P, can be written as: $\small{(lr,mr,nr)}$
7. $\small{lr,mr~\text{and}~nr}$ are mere numbers.
• They are called direction ratios of $\small{\vec{r}}$. We will see more details about this in later sections.
In the next section, we will see types of vectors.
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