Maths

Given,  $P\left(1,2,3\right)\&Q\left(4,5,6\right)$

So,

$\begin{array}{l}\overrightarrow{PQ}=(4-1)\hat{i}+(5-2)\hat{j}+(6-3)\hat{k}\\ =3\hat{i}+3\hat{j}+3\hat{k}\end{array}$

Kindly go through the solution

The given vectors are

$\overrightarrow{a}=\hat{i}-2\hat{j}+\hat{k}$

$\overrightarrow{b}=\widehat{-2i}+4\hat{j}+5\hat{k}$

$\overrightarrow{c}=\hat{i}-6\hat{j}-7\hat{k}$

The sum of the vector is

$\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\left(a_1+a_2+a_3\right)\hat{i}+\left(b_1+b_2+b_3\right)\widehat{\hat{j}+\left(c_1+c_2+c_3\right)\hat{k}}$

$\begin{array}{l}=(1-2+1)\hat{i}+(-2+4-6)\hat{j}+(1+5-7)\hat{k}\\ =0.\hat{i}+(-4)\hat{j}+(-1)\hat{k}\\ =-4\hat{j}-\stackrel{}{k}\end{array}$

43. Given system of inequality is

2x+y≥ 8- (1)

x+2y≥ 10- (2)

The corresponding equations are

2x + y = 8

and x + 2y = 10

Now, putting (x, y)= (0,0) in inequality (1) and (2),

2 * 0+8 ≥ 8

0 ≥ 8 which is not true.

and 0+2 * 0 ≥ 10

0 ≥ 10 which is not true.

So, solution of plane of inequality (1) and (2) does not include the origin (0,0)

? The required solution of the given system of inequality is the shaded region.

Let the vector with initial point P (2,1) and terminal point Q. (-5,7) can be shown as,

$\begin{array}{l}\overrightarrow{PQ}=(-5,-2)\hat{i}+(7,-1)\hat{j}\\ \overrightarrow{PQ}=-7\hat{i}+6\hat{j}\end{array}$

The scalar components are -7 and 6.

The vector components are -7i and 6j.

Note that two vector are equal only if their corresponding components are equal.

Thus, the given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ will be equal if and only if $x=2\&y=3$

42. The given system of inequality is

x+y≤ 6 - (1)

x+y≥ 4- (2)

So the corresponding equations are

x+y=6

and x + y = 4

Putting (x, y)= (0,0) in equality (1) and (2),

0+0 ≤ 6 and 0 + 0 ≥ 4

0 ≤ 6 is true.  => 0 ≥ 4 is false.

So, solution of plane of inequality (1) includes the origin and inequality (2) does not includes the origin.

? The reqd solution of the given system of inequality is the shaded region.

Two different vectors having same magnitude: -

(i) $2\hat{i}+\hat{j}+3\hat{k}$

(ii) $\hat{i}+3\hat{j}+2\hat{k}$

Kindly go through the solution