Higher Secondary Mathematics: Free Vectors

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Tuesday, May 12, 2026

26.5 - Solved Examples

In the previous section, we saw components of a vector. We saw a solved example also. In this section, we will see a few more solved examples.

Solved example 26.7
Find the values of x and y so that the vectors $\small{2\hat{i} + 3\hat{j}}$ and $\small{x\hat{i} + y\hat{j}}$ are equal
Solution:
1. If two vectors are equal, their corresponding components will be equal.
2. Let us equate the corresponding components:
• Equating the vector components along the x-axis, we get: $\small{2\hat{i} = x\hat{i}}$. Therefore, x = 2
• Equating the vector components along the y-axis, we get: $\small{3\hat{j} = y\hat{i}}$. Therefore, y = 3

Solved example 26.8
Let $\small{\vec{a} = \hat{i} + 2\hat{j}}$ and $\small{\vec{b} = 2\hat{i} + \hat{j}}$.
Is $\small{\left|\vec{a} \right| = \left|\vec{b} \right|}$?
Are the vectors $\small{\vec{a}~\text{and}~\vec{b}}$ equal?
Solution:
Part (a): Comparing the magnitudes
1. $\small{\left|\vec{a} \right| = \sqrt{1^2 + 2^2} = \sqrt{5}}$
2. $\small{\left|\vec{b} \right| = \sqrt{2^2 + 1^2} = \sqrt{5}}$
3. The magnitudes are equal. So $\small{\left|\vec{a} \right| = \left|\vec{b} \right|}$

Part (b): Checking equality of vectors
1. If two vectors are equal, their corresponding components will be equal.
2. Let us compare the corresponding components:
• Comparing the vector components along the x-axis, we see that: $\small{\hat{i} ~\ne~ 2\hat{i}}$.
• Comparing the vector components along the y-axis, we see that: $\small{2\hat{j} ~\ne~ \hat{j}}$
3. Two vectors cannot be equal if even one of the corresponding components are not equal. So the given two vectors are not equal.

Solved example 26.9
Compute the magnitude of the following vectors:
$\small{\vec{a} = \hat{i} + \hat{j} + \hat{k}}$
$\small{\vec{b} = 2\hat{i} - 7\hat{j} - 3\hat{k}}$
$\small{\vec{c} = \frac{1}{\sqrt{3}} \hat{i} + \frac{1}{\sqrt{3}} \hat{j} - \frac{1}{\sqrt{3}} \hat{k}}$
Solution:
Part (a):
$\small{\left|\vec{a} \right| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}}$
Part (b):
$\small{\left|\vec{b} \right| = \sqrt{2^2 + (-7)^2 + (-3)^2} = \sqrt{4 + 49 + 9} = \sqrt{62}}$
Part (c):
$\small{\left|\vec{c} \right| = \sqrt{\left(\frac{1}{\sqrt{3}} \right)^2 + \left(\frac{1}{\sqrt{3}} \right)^2 + \left(-\frac{1}{\sqrt{3}} \right)^2} =\frac{1}{3} + \frac{1}{3} + \frac{1}{3} = \frac{3}{3} = 1}$

Solved example 26.10
Write two different vectors having same magnitude
Solution:
We want two different vectors having the same magnitude. It can be done in 2 steps:
1. First write any convenient vector, say: $\small{\vec{a} = 3\hat{i}+4\hat{j}}$
2. Now we write the second vector by changing the order/sign of the scalar components. So the two vectors are:
$\small{\vec{a} = 3\hat{i}+4\hat{j}}$
$\small{\vec{b} = 4\hat{i}+3\hat{j}}$
• We have the magnitudes:
$\small{\left|\vec{a} \right| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5}$
$\small{\left|\vec{b} \right| = \sqrt{4^2 + 3^2} = \sqrt{25} = 5}$
• Fig.26.20 below shows the difference in directions:

Fig.26.20

◼ Note:
There are other possibilities also:
$\small{\vec{a} = 3\hat{i}+4\hat{j}}$
$\small{\vec{c} = 3\hat{i}-4\hat{j}}$
$\small{\vec{d} = -4\hat{i}+3\hat{j}}$
$\small{\vec{e} = -3\hat{i}-4\hat{j}}$
• All these vectors have different directions. But they have the same magnitude. We can pick any two from them.

Solved example 26.11
Find the unit vector in the direction of the vector $\small{\vec{a} = 2\hat{i}+3\hat{j}+\hat{k}}$
Solution:
• We have: $\small{\hat{a}=\frac{\vec{a}}{\left|\vec{a} \right|}}$
• Thus we get:
$\small{\begin{array}{ll} {~\color{magenta}    1    }    &{{}}    &{\hat{a}}    & {~=~}    &{\frac{2\hat{i}+3\hat{j}+\hat{k}}{\sqrt{2^2 + 3^2 + 1^2}}}
\\ {~\color{magenta}    2    }    &{}    &{}    & {~=~}    &{\frac{2\hat{i}+3\hat{j}+\hat{k}}{\sqrt{14}}}
\\ {~\color{magenta}    3    }    &{}    &{}    & {~=~}    &{\left(\frac{2}{\sqrt{14}} \right) \hat{i}+\left(\frac{3}{\sqrt{14}} \right) \hat{j}+\left(\frac{1}{\sqrt{14}} \right) \hat{k}}
\\ \end{array}}$

Solved example 26.12
Find the unit vector in the direction of the vector $\small{\vec{a} = \hat{i}+\hat{j}+2\hat{k}}$
Solution:
• We have: $\small{\hat{a}=\frac{\vec{a}}{\left|\vec{a} \right|}}$
• Thus we get:
$\small{\begin{array}{ll} {~\color{magenta}    1    }    &{{}}    &{\hat{a}}    & {~=~}    &{\frac{\hat{i}+\hat{j}+2\hat{k}}{\sqrt{1^2 + 1^2 + 2^2}}}
\\ {~\color{magenta}    2    }    &{}    &{}    & {~=~}    &{\frac{\hat{i}+\hat{j}+2\hat{k}}{\sqrt{6}}}
\\ {~\color{magenta}    3    }    &{}    &{}    & {~=~}    &{\left(\frac{1}{\sqrt{6}} \right) \hat{i}+\left(\frac{1}{\sqrt{6}} \right) \hat{j}+\left(\frac{2}{\sqrt{6}} \right) \hat{k}}
\\ \end{array}}$

Solved example 26.13
Find a vector in the direction of the vector $\small{\vec{a} = \hat{i}-2\hat{j}}$ that has magnitude 7 units.
Solution:
1. First we will write the unit vector $\small{\hat{a}}$
• We have: $\small{\hat{a}=\frac{\vec{a}}{\left|\vec{a} \right|}}$
• Thus we get:
$\small{\begin{array}{ll} {~\color{magenta}    1    }    &{{}}    &{\hat{a}}    & {~=~}    &{\frac{\hat{i}-2\hat{j}}{\sqrt{1^2 + (-2)^2}}}
\\ {~\color{magenta}    2    }    &{}    &{}    & {~=~}    &{\frac{\hat{i}-2\hat{j}}{\sqrt{5}}}
\\ {~\color{magenta}    3    }    &{}    &{}    & {~=~}    &{\left(\frac{1}{\sqrt{5}} \right) \hat{i}-\left(\frac{2}{\sqrt{5}} \right) \hat{j}}
\\ \end{array}}$
• This unit vector has the same direction as $\small{\vec{a}}$

2. So the required vector is:
$\small{\begin{array}{ll} {~\color{magenta} 1 } &{{}} &{7\hat{a}} & {~=~} &{7\left[\left(\frac{1}{\sqrt{5}} \right) \hat{i}-\left(\frac{2}{\sqrt{5}} \right) \hat{j} \right]}
\\ {~\color{magenta} 2 } &{} &{} & {~=~} &{\left(\frac{7}{\sqrt{5}} \right) \hat{i}-\left(\frac{14}{\sqrt{5}} \right) \hat{j}}
\\ \end{array}}$

Solved example 26.14
Find a vector in the direction of the vector $\small{\vec{a} = 5\hat{i}-\hat{j}+2\hat{k}}$ that has magnitude 8 units.
Solution:
1. First we will write the unit vector $\small{\hat{a}}$
• We have: $\small{\hat{a}=\frac{\vec{a}}{\left|\vec{a} \right|}}$
• Thus we get:
$\small{\begin{array}{ll} {~\color{magenta}    1    }    &{{}}    &{\hat{a}}    & {~=~}    &{\frac{5\hat{i}-\hat{j}+2\hat{k}}{\sqrt{5^2 + (-1)^2 + 2^2}}}
\\ {~\color{magenta}    2    }    &{}    &{}    & {~=~}    &{\frac{5\hat{i}-\hat{j}+2\hat{k}}{\sqrt{30}}}
\\ {~\color{magenta}    3    }    &{}    &{}    & {~=~}    &{\left(\frac{5}{\sqrt{30}} \right) \hat{i}-\left(\frac{1}{\sqrt{30}} \right) \hat{j}+\left(\frac{2}{\sqrt{30}} \right) \hat{j}}
\\ \end{array}}$

• This unit vector has the same direction as $\small{\vec{a}}$

2. So the required vector is:
$\small{\begin{array}{ll} {~\color{magenta}    1    }    &{{}}    &{8\hat{a}}    & {~=~}    &{8\left[\left(\frac{5}{\sqrt{30}} \right) \hat{i}-\left(\frac{1}{\sqrt{30}} \right) \hat{j}+\left(\frac{2}{\sqrt{30}} \right) \hat{j} \right]}
\\ {~\color{magenta}    2    }    &{}    &{}    & {~=~}    &{\left(\frac{40}{\sqrt{30}} \right) \hat{i}-\left(\frac{8}{\sqrt{30}} \right) \hat{j}+\left(\frac{16}{\sqrt{30}} \right) \hat{j}}
\\ \end{array}}$

Solved example 26.15
Write two different vectors having same direction
Solution:
We want two different vectors having the same direction. It can be done in 6 steps:
1. First write any convenient vector, say: $\small{\vec{a} = 2\hat{i}+3\hat{j}}$
2. Now write $\small{\hat{a}}$. We get:
$\small{\hat{a}=\left(\frac{2}{\sqrt{13}} \right) \hat{i}+\left(\frac{3}{\sqrt{13}} \right) \hat{j}}$
• The reader may write all steps involved in finding the unit vector
3. Multiply the unit vector in (2), by any convenient scalar, to get a new vector $\small{\vec{b}}$:
$\small{\begin{array}{ll} {~\color{magenta}    1    }    &{{}}    &{\vec{b}}    & {~=~}    &{\sqrt{13}\,\hat{a}}
\\ {~\color{magenta}    2    }    &{}    &{}    & {~=~}    &{\sqrt{13}\left[\left(\frac{2}{\sqrt{13}} \right) \hat{i}+\left(\frac{3}{\sqrt{13}} \right) \hat{j} \right]}
\\ {~\color{magenta}    3    }    &{\Rightarrow}    &{\vec{b}}    & {~=~}    &{2 \hat{i}+3 \hat{j}}
\\ \end{array}}$
• Note that: $\small{\vec{b}=\vec{a}}$
4. Multiply the unit vector in (2), by any other convenient scalar, to get a third vector $\small{\vec{c}}$:
$\small{\begin{array}{ll} {~\color{magenta}    1    }    &{{}}    &{\vec{c}}    & {~=~}    &{2\sqrt{13}\,\hat{a}}
\\ {~\color{magenta}    2    }    &{}    &{}    & {~=~}    &{2\sqrt{13}\left[\left(\frac{2}{\sqrt{13}} \right) \hat{i}+\left(\frac{3}{\sqrt{13}} \right) \hat{j} \right]}
\\ {~\color{magenta}    3    }    &{\Rightarrow}    &{\vec{c}}    & {~=~}    &{4 \hat{i}+6 \hat{j}}
\\ \end{array}}$
5. $\small{\vec{b}~\text{and}~\vec{c}}$ are the required vectors
6. Fig.26.21 shows the two vectors:

Fig.26.21

Solved example 26.16
Find the sum of the vectors
$\small{\vec{a} = \hat{i}-2 \hat{j} + \hat{k}}$
$\small{\vec{b} = -2\hat{i}+4 \hat{j} + 5\hat{k}}$
$\small{\vec{c} = \hat{i}-6 \hat{j} -7 \hat{k}}$
Solution:
$\small{\begin{array}{ll} {~\color{magenta}    1    }    &{{}}    &{\vec{a}+\vec{b}+\vec{c}}    & {~=~}    &{(1-2+1)\hat{i}+(-2+4-6) \hat{j} + (1+5-7)\hat{k}}
\\ {~\color{magenta}    2    }    &{}    &{}    & {~=~}    &{(0)\hat{i}+(-4) \hat{j} + (-1)\hat{k}}
\\ {~\color{magenta}    3    }    &{}    &{}    & {~=~}    &{-4 \hat{j}-\hat{k}}
\\ \end{array}}$

Solved example 26.17
Find unit vector in the direction of the sum of the vectors
$\small{\vec{a} = 2\hat{i}+2 \hat{j} -5 \hat{k}}$
$\small{\vec{b} = 2\hat{i}+ \hat{j} + 3\hat{k}}$
Solution:
1. First we will find the sum:
$\small{\begin{array}{ll} {~\color{magenta}    1    }    &{{}}    &{\vec{c} = \vec{a}+\vec{b}}    & {~=~}    &{(2+2)\hat{i}+(2+1) \hat{j} + (-5+3)\hat{k}}
\\ {~\color{magenta}    2    }    &{}    &{}    & {~=~}    &{(4)\hat{i}+(3) \hat{j} + (-2)\hat{k}}
\\ {~\color{magenta}    3    }    &{}    &{}    & {~=~}    &{4\hat{i}+3 \hat{j} -2\hat{k}}
\\ \end{array}}$

Solved example 26.18
Find unit vector in the direction of the sum of the vectors
$\small{\vec{a} = 2\hat{i}- \hat{j} +2 \hat{k}}$
$\small{\vec{b} = -\hat{i}+ \hat{j} - \hat{k}}$
Solution:
1. First we will find the sum:
$\small{\begin{array}{ll} {~\color{magenta}    1    }    &{{}}    &{\vec{c} = \vec{a}+\vec{b}}    & {~=~}    &{(2-1)\hat{i}+(-1+1) \hat{j} + (2-1)\hat{k}}
\\ {~\color{magenta}    2    }    &{}    &{}    & {~=~}    &{(1)\hat{i}+(0) \hat{j} + (1)\hat{k}}
\\ {~\color{magenta}    3    }    &{}    &{}    & {~=~}    &{\hat{i}+\hat{k}}
\\ \end{array}}$

Solved example 26.19
Write the direction ratios of the vector $\small{\vec{a} = \hat{i} + \hat{j} -2 \hat{k}}$ and hence calculate its direction cosines.
Solution:
1. Any vector is the resultant of three component vectors:
    ♦ Component along the OX axis, which is $\small{x \hat{i}}$
    ♦ Component along the OY axis, which is $\small{y \hat{j}}$
    ♦ Component along the OZ axis, which is $\small{z \hat{k}}$
• So any given vector can be written as: $\small{x\hat{i}+y\hat{j}+z\hat{k}}$
2. The vector given to us is: $\small{\vec{a}=\hat{i}+\hat{j}-2\hat{k}}$
• Comparing the corresponding components, we get: x = 1, y = 1 and z = −2
3. $\small{r = \left|\vec{a} \right|=\sqrt{1^2 + 1^2 + (-2)^2} = \sqrt{6}}$
4. The three direction ratios are:
    ♦ $\small{lr = x = 1}$
    ♦ $\small{mr = y = 1}$
    ♦ $\small{nr = z = -2}$
Where $\small{l,~m,~n}$ are the direction cosines
5. So the three direction cosines are:
    ♦ $\small{l = \frac{x}{r} = \frac{1}{\sqrt{6}}}$
    ♦ $\small{m = \frac{y}{r} = \frac{1}{\sqrt{6}}}$
    ♦ $\small{n = \frac{z}{r} = \frac{-2}{\sqrt{6}}}$

Solved example 26.20
Find the direction cosines of the vector $\small{\vec{a} = \hat{i} + 2\hat{j} +3 \hat{k}}$.
Solution:
1. Any vector is the resultant of three component vectors:
    ♦ Component along the OX axis, which is $\small{x \hat{i}}$
    ♦ Component along the OY axis, which is $\small{y \hat{j}}$
    ♦ Component along the OZ axis, which is $\small{z \hat{k}}$
• So any given vector can be written as: $\small{x\hat{i}+y\hat{j}+z\hat{k}}$
2. The vector given to us is: $\small{\vec{a}=\hat{i}+2\hat{j}+3\hat{k}}$
• Comparing the corresponding components, we get: x = 1, y = 2 and z = 3
3. $\small{r = \left|\vec{a} \right|=\sqrt{1^2 + 2^2 + 3^2} = \sqrt{14}}$
4. The three direction ratios are:
    ♦ $\small{lr = x = 1}$
    ♦ $\small{mr = y = 2}$
    ♦ $\small{nr = z = 3}$
Where $\small{l,~m,~n}$ are the direction cosines
5. So the three direction cosines are:
    ♦ $\small{l = \frac{x}{r} = \frac{1}{\sqrt{14}}}$
    ♦ $\small{m = \frac{y}{r} = \frac{2}{\sqrt{14}}}$
    ♦ $\small{n = \frac{z}{r} = \frac{3}{\sqrt{14}}}$

Solved example 26.21
Show that the vector $\small{\vec{a} = \hat{i} + \hat{j} + \hat{k}}$ is equally inclined to the axes OX, OY and OZ.
Solution:
1. Any vector is the resultant of three component vectors:
    ♦ Component along the OX axis, which is $\small{x \hat{i}}$
    ♦ Component along the OY axis, which is $\small{y \hat{j}}$
    ♦ Component along the OZ axis, which is $\small{z \hat{k}}$
• So any given vector can be written as: $\small{x\hat{i}+y\hat{j}+z\hat{k}}$
2. The vector given to us is: $\small{\vec{a}=\hat{i}+\hat{j}+\hat{k}}$
• Comparing the corresponding components, we get: x = 1, y = 1 and z = 1
3. $\small{r = \left|\vec{a} \right|=\sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}}$
4. The three direction ratios are:
    ♦ $\small{lr = x = 1}$
    ♦ $\small{mr = y = 1}$
    ♦ $\small{nr = z = 1}$
Where $\small{l,~m,~n}$ are the direction cosines
5. So the three direction cosines are:
    ♦ $\small{l = \frac{x}{r} = \frac{1}{\sqrt{3}}}$
    ♦ $\small{m = \frac{y}{r} = \frac{1}{\sqrt{3}}}$
    ♦ $\small{n = \frac{z}{r} = \frac{1}{\sqrt{3}}}$
6. The three direction cosines are equal. That means, the three angles are equal.

Solved example 26.22
Show that the vectors $\small{\vec{a}=2\hat{i}-3\hat{j}+4\hat{k}}$ and $\small{\vec{b}=-4\hat{i}+6\hat{j}-8\hat{k}}$ are collinear.
Solution:
1. Unit vector in the direction of $\small{\vec{a}}$
• We have: $\small{\hat{a}=\frac{\vec{a}}{\left|\vec{a} \right|}}$
• Thus we get:
$\small{\begin{array}{ll} {~\color{magenta}    1    }    &{{}}    &{\hat{a}}    & {~=~}    &{\frac{2\hat{i}-3\hat{j}+4\hat{k}}{\sqrt{2^2 + (-3)^2 + (4)^2}}}
\\ {~\color{magenta}    2    }    &{}    &{}    & {~=~}    &{\frac{2\hat{i}-3\hat{j}+4\hat{k}}{\sqrt{29}}}
\\ {~\color{magenta}    3    }    &{}    &{}    & {~=~}    &{\left(\frac{2}{\sqrt{29}} \right) \hat{i}-\left(\frac{3}{\sqrt{29}} \right) \hat{j}+\left(\frac{4}{\sqrt{29}} \right) \hat{k}}
\\ \end{array}}$

2. Unit vector in the direction of $\small{\vec{b}}$
• We have: $\small{\hat{b}=\frac{\vec{b}}{\left|\vec{b} \right|}}$
• Thus we get:
$\small{\begin{array}{ll} {~\color{magenta}    1    }    &{{}}    &{\hat{b}}    & {~=~}    &{\frac{-4\hat{i}+6\hat{j}-8\hat{k}}{\sqrt{(-4)^2 + 6^2 + (-8)^2}}}
\\ {~\color{magenta}    2    }    &{}    &{}    & {~=~}    &{\frac{-4\hat{i}+6\hat{j}-8\hat{k}}{\sqrt{116}}~=~\frac{-4\hat{i}+6\hat{j}-8\hat{k}}{2\sqrt{29}}}
\\ {~\color{magenta}    3    }    &{}    &{}    & {~=~}    &{\left(\frac{-2}{\sqrt{29}} \right) \hat{i}+\left(\frac{3}{\sqrt{29}} \right) \hat{j}-\left(\frac{4}{\sqrt{29}} \right) \hat{k}}
\\ \end{array}}$

3. We see that, $\small{\hat{a}=-\hat{b}}$
• That means: $\small{\hat{a}}$ has the exact opposite direction of $\small{\hat{b}}$
• That means: $\small{\vec{a}}$ has the exact opposite direction of $\small{\vec{b}}$
• That means: $\small{\vec{a}}$ and $\small{\vec{b}}$ are parallel.
• Therefore $\small{\vec{a}}$ and $\small{\vec{b}}$ are collinear.

Solved example 26.23
If $\small{\vec{a}~\text{and}~\vec{b}}$ are two collinear vectors, then which of the following are incorrect:
(a) $\small{\vec{b} = \lambda\vec{a}}$ for some scalar $\small{\lambda}$
(b) $\small{\vec{a} = \pm \vec{b}}$
(c) the respective components of $\small{\vec{a}~\text{and}~\vec{b}}$ are proportional
(d) both the vectors $\small{\vec{a}~\text{and}~\vec{b}}$ have same direction, but different magnitudes
Solution:
Part (a):
Given that $\small{\vec{a}~\text{and}~\vec{b}}$ are collinear. That means, they are parallel. So (a) is true.
Part (b):
$\small{\vec{a}~\text{and}~\vec{b}}$ are parallel. But they need not have the same direction. The directions may be opposite to each other. So (b) is true.
Part (c):
Since (a) is true, we can multiply each component of $\small{\vec{a}}$ by $\small{\lambda}$. That means, corresponding components are proportional. So (c) is true.
Part (d):
$\small{\vec{a}~\text{and}~\vec{b}}$ are parallel. But they need not have the same direction. The directions may be opposite to each other. So (d) is false.

Therefore, the correct option is (d)

Solved example 26.24
Find the value of x for which $\small{x\left(\hat{i}+\hat{j}+\hat{k} \right)}$ is a unit vector
Solution:
1. Let $\small{\vec{a}=x\left(\hat{i}+\hat{j}+\hat{k} \right)}$.
Then we can write: $\small{\vec{a}=x\hat{i}+x\hat{j}+x\hat{k}}$
2. Now we can calculate the magnitude:
$\small{\left|\vec{a} \right| = \sqrt{x^2 + x^2 + x^2} = \sqrt{3x^2} = \pm\sqrt{3}\,x}$
3. $\small{\vec{a}}$ is to be a unit vector. So it's magnitude must be 1. We can write:
$\small{\begin{array}{ll} {~\color{magenta}    1    }    &{{}}    &{\left|\vec{a} \right|}    & {~=~}    &{1}
\\ {~\color{magenta}    2    }    &{\Rightarrow}    &{\pm\sqrt{3}\,x}    & {~=~}    &{1}
\\ {~\color{magenta}    3    }    &{\Rightarrow}    &{x}    & {~=~}    &{\pm\frac{1}{\sqrt{3}}}
\\ \end{array}}$

Solved example 26.25
Find a vector of magnitude 5 units, and parallel to the resultant of the vectors
$\small{\vec{a}=2\hat{i}+3\hat{j}-\hat{k}~\text{and}~\vec{b}=\hat{i}-2\hat{j}+\hat{k}}$
Solution:
1. First we find the resultant $\small{\vec{a}+\vec{b}}$:
$\small{\begin{array}{ll} {~\color{magenta}    1    }    &{{}}    &{\vec{c}}    & {~=~}    &{\vec{a}+\vec{b}}
\\ {~\color{magenta}    2    }    &{}    &{}    & {~=~}    &{2\hat{i}+3\hat{j}-\hat{k}~+~\hat{i}-2\hat{j}+\hat{k}}
\\ {~\color{magenta}    3    }    &{}    &{}    & {~=~}    &{3\hat{i}+\hat{j}}
\\ \end{array}}$

2. Now we will write the unit vector $\small{\hat{c}}$
• We have: $\small{\hat{c}=\frac{\vec{c}}{\left|\vec{c} \right|}}$
• Thus we get:
$\small{\begin{array}{ll} {~\color{magenta}    1    }    &{{}}    &{\hat{c}}    & {~=~}    &{\frac{3\hat{i}+\hat{j}}{\sqrt{3^2 + 1^2}}}
\\ {~\color{magenta}    2    }    &{}    &{}    & {~=~}    &{\frac{3\hat{i}+\hat{j}}{\sqrt{10}}}
\\ {~\color{magenta}    3    }    &{}    &{}    & {~=~}    &{\left(\frac{3}{\sqrt{10}} \right) \hat{i}+\left(\frac{1}{\sqrt{10}} \right) \hat{j}}
\\ \end{array}}$
• This unit vector has the same direction as $\small{\vec{c}}$

3. So the required vector is:
$\small{\begin{array}{ll} {~\color{magenta}    1    }    &{{}}    &{5\hat{c}}    & {~=~}    &{5\left[\left(\frac{3}{\sqrt{10}} \right) \hat{i}+\left(\frac{1}{\sqrt{10}} \right) \hat{j} \right]}
\\ {~\color{magenta}    2    }    &{}    &{}    & {~=~}    &{\left(\frac{15}{\sqrt{10}} \right) \hat{i}+\left(\frac{5}{\sqrt{10}} \right) \hat{j}}
\\ {~\color{magenta}    3    }    &{}    &{}    & {~=~}    &{\left(\frac{15 \sqrt{10}}{10} \right) \hat{i}+\left(\frac{5 \sqrt{10}}{10} \right) \hat{j}}
\\ {~\color{magenta}    4    }    &{}    &{}    & {~=~}    &{\left(\frac{3 \sqrt{10}}{2} \right) \hat{i}+\left(\frac{\sqrt{10}}{2} \right) \hat{j}}
\\ \end{array}}$

Solved example 26.26
If
$\small{\vec{a}=\hat{i}+\hat{j}+\hat{k}}$
$\small{\vec{b}=2\hat{i}-\hat{j}+3\hat{k}}$
$\small{\vec{c}=\hat{i}-2\hat{j}+\hat{k}}$
find a unit vector parallel to the vector $\small{2\vec{a}-\vec{b}+3\vec{c}}$
Solution:
1. First we find the vector $\small{\vec{d}=2\vec{a}-\vec{b}+3\vec{c}}$.
We get: $\small{\vec{d}=3\hat{i}-3\hat{j}+2\hat{k}}$
2. Now we will write the unit vector $\small{\hat{d}}$
• We have: $\small{\hat{d}=\frac{\vec{d}}{\left|\vec{d} \right|}}$
• Thus we get:
$\small{\begin{array}{ll} {~\color{magenta}    1    }    &{{}}    &{\hat{d}}    & {~=~}    &{\frac{3\hat{i}-3\hat{j}+2\hat{k}}{\sqrt{3^2 +(-3)^2 + 2^2}}}
\\ {~\color{magenta}    2    }    &{}    &{}    & {~=~}    &{\frac{3\hat{i}-3\hat{j}+2\hat{k}}{\sqrt{22}}}
\\ {~\color{magenta}    3    }    &{}    &{}    & {~=~}    &{\left(\frac{3}{\sqrt{22}} \right) \hat{i}-\left(\frac{3}{\sqrt{22}} \right) \hat{j}+\left(\frac{2}{\sqrt{22}} \right) \hat{k}}
\\ \end{array}}$
• This unit vector has the same direction as $\small{\vec{d}}$.

Solved example 26.27
Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are $\small{\frac{1}{\sqrt{3}},~\frac{1}{\sqrt{3}},~\frac{1}{\sqrt{3}}}$
Solution:
1. Any vector is the resultant of three component vectors:
    ♦ Component along the OX axis, which is $\small{x \hat{i}}$
    ♦ Component along the OY axis, which is $\small{y \hat{j}}$
    ♦ Component along the OZ axis, which is $\small{z \hat{k}}$
• So any given vector can be written as: $\small{\vec{a}=x\hat{i}+y\hat{j}+z\hat{k}}$
2. Let $\small{r = \left|\vec{a} \right|=\sqrt{x^2 + y^2 + z^2}}$
3. The three direction ratios are:
    ♦ $\small{lr = x}$
    ♦ $\small{mr = y}$
    ♦ $\small{nr = z}$
Where $\small{l,~m,~n}$ are the direction cosines
4. So the three direction cosines are:
    ♦ $\small{l = \frac{x}{r}}$
    ♦ $\small{m = \frac{y}{r}}$
    ♦ $\small{n = \frac{z}{r}}$
5. The vector is equally inclined to the axes OX, OY and OZ. So we can write:
$\small{l=m=n}$
• Then from (4), we get:
$\small{\Rightarrow x=y=z}$
6. So from (2), we get:
• $\small{r=\sqrt{x^2 + x^2 + x^2}=\sqrt{3x^2}=\sqrt{3}\,x }$
• $\small{r=\sqrt{y^2 + y^2 + y^2}=\sqrt{3y^2}=\sqrt{3}\,y }$
• $\small{r=\sqrt{z^2 + z^2 + z^2}=\sqrt{3z^2}=\sqrt{3}\,z }$
• Note: Here we take only the +ve roots because, r is a distance.
7. Based on (4) and (6), we get:
• $\small{l = \frac{x}{r}=\frac{x}{\sqrt{3}\,x}=\frac{1}{\sqrt{3}}}$
• $\small{m = \frac{y}{r}=\frac{y}{\sqrt{3}\,y}=\frac{1}{\sqrt{3}}}$
• $\small{n = \frac{z}{r}=\frac{z}{\sqrt{3}\,z}=\frac{1}{\sqrt{3}}}$

In the next section, we will see vector joining two points.

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Sunday, April 26, 2026

26.1 - Types of Vectors

In the previous section, we saw direction cosines. In this section, we will see types of vectors.

We need to learn about seven types of vectors:
1. Zero Vector
For a vector, if the terminal point is same as the initial point, then that vector is called a zero vector.
• A zero vector is denoted as $\small{\vec{0}}$
• A zero vector is also known as a null vector.
• A zero vector cannot be assigned any definite direction because, it has no magnitude.
• Alternatively, a zero vector can be regarded as a vector having any of the infinite possible directions.
• Vectors $\small{\vec{AA}, \vec{BB},~.~.~}$ are examples of zero vector.

2. Unit Vector
If the magnitude of a vector is 1 unit, then it is called a unit vector.
• Suppose that, we are given a vector $\small{\vec{a}}$. We can draw a unit vector which has the same direction as $\small{\vec{a}}$. Such a vector is denoted as $\small{\hat{a}}$

3. Coinitial Vectors
If two or more vectors have the same initial point, then they are called coinitial vectors.
• $\small{\vec{AB}}$ and $\small{\vec{AC}}$ are coinitial vectors

4. Collinear Vectors
If two or more vectors are parallel to the same line, then they are called collinear vectors.
• Collinear vectors need not lie along the same line.
• Collinear vectors can have only two possible directions. One direction exactly opposite to the other.

5. Equal Vectors
Two vectors are said to be equal if they have the same magnitude and direction.
• Equal vectors need not have the same initial point.
• If two vectors $\small{\vec{a}~\text{and}~\vec{b}}$ are equal, then we write $\small{\vec{a} = \vec{b}}$

6. Negative of a Vector
A vector is said to be the negative of another vector, if they have the same magnitude but opposite directions.
• $\small{\vec{AB}}$ is the negative of $\small{\vec{BA}}$. We write: $\small{\vec{BA} = -\vec{AB}}$

7. Free Vectors
Free vectors are those which can be moved around. The only condition is that, the magnitude and direction should not be altered.
• In this chapter, we will be dealing with free vectors only.

Now we will see some solved examples

Solved example 26.1
Represent graphically a displacement of 40 km, 30^o west of south.
Solution:
1. An object undergoes a displacement of 40 km in a specified direction. We are asked to draw the displacement vector.

2. We setup the rectangular coordinate system in such a way that, the origin O of the coordinate system coincides with the initial position of the object. Also, the following directions should be clearly marked:
    ♦ +ve x direction towards east
    ♦ +ve y direction towards north
    ♦ −ve x direction towards west
    ♦ −ve y direction towards south
This is shown in fig.26.8 below:

Fig.26.8

3. Next step is to draw a dashed line which satisfies two conditions:
♦ The line must make 30^o with the south direction
♦ The line must be inclined towards west
This dashed line is shown in fig.a

4. Next step is to mark a point P on the dashed line in such a way that, OP represents 40 km. A suitable scale can be used based on the size of the paper available.

5. Now, the dashed portion between O and P should be changed to a continuous line. And an arrowhead can be placed at P. This is shown in fig.b
• Thus we get $\small{\vec{OP}}$ which is the displacement vector.

6. Scale of the drawing should be written on the top left corner.

Solved example 26.2
Classify the following measures as scalars and vectors.
(i) 5 seconds (ii) 1000 cm³ (iii) 10 newton (iv) 30 km/hr (v) 10 g/cm³ (vi) 20 m/s towards north
Solution:
(i) Time has magnitude only. It has no direction. So 5 seconds is a scalar quantity
(ii) Volume has magnitude only. It has no direction. So 1000 cm³ is a scalar quantity
(iii) Force has both magnitude and direction. So 10 newton is a vector quantity.
(iv) Speed has magnitude only. It has no direction. So 30 km/hr is a scalar quantity.
(v) Density has magnitude only. It has no direction. So 10 g/cm³ is a scalar quantity.
(vi) Velocity has both magnitude and direction. So "20 m/s towards north" is a vector quantity.

Solved example 26.3
In fig.26.9, which of the vectors are:
(i) Collinear (ii) Equal (iii) Coinitial

Checking whether the given vectors are equal or collinear

Fig.26.9

Solution:
1. We are given four vectors.
• Each of $\small{\vec{a}}$ and $\small{\vec{c}}$ has a small white perpendicular line. This line divides the vector into two equal parts. So these two vectors have a magnitude of two units.
• Each of $\small{\vec{b}}$ and $\small{\vec{d}}$ has two small white perpendicular lines. These lines divide the vectors into three equal parts. So these two vectors have a magnitude of three units.
2. We see that $\small{\vec{a}~\text{and}~\vec{c}}$ have the same direction. Also, both have a magnitude of 2 units. So they are equal vectors.
3. $\small{\vec{b}, \vec{c}~\text{and}~\vec{d}}$ have the same initial point. So they are coinitial vectors.
4. $\small{\vec{a}, \vec{c}~\text{and}~\vec{d}}$ are parallel. So they are collinear vectors

Solved example 26.4
Represent graphically a displacement of 40 km, 30^o east of north
Solution:

1. An object undergoes a displacement of 40 km in a specified direction. We are asked to draw the displacement vector.

Fig.26.10

3. Next step is to draw a dashed line which satisfies two conditions:
♦ The line must make 30^o with the north direction
♦ The line must be inclined towards east
This dashed line is shown in fig.a

4. Next step is to mark a point P on the dashed line in such a way that, OP represents 40 km. A suitable scale can be used based on the size of the paper available.

6. Scale of the drawing should be written on the top left corner.

Solved example 26.5
In fig.26.11 (a square), identify the following vectors.
(i) Coinitial (ii) Equal (iii) Collinear but not equal

Fig.26.11

Solution:
1. We are given four vectors. They form the sides of a square. So all four of them have the same magnitude.
2. $\small{\vec{a}~\text{and}~\vec{d}}$ have the same initial point. So they are coinitial vectors.
3. $\small{\vec{b}~\text{and}~\vec{d}}$ have the same magnitude and direction. So they are equal vectors.
4. $\small{\vec{a}~\text{and}~\vec{c}}$ have the same magnitude but opposite directions. So they are not equal vectors. However, since they are parallel, they are collinear.

The link below gives a few more solved examples:

Exercise 26.1

In the next section, we will see addition of vectors.

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Tuesday, May 12, 2026

26.5 - Solved Examples

Sunday, April 26, 2026

26.1 - Types of Vectors