====== Question 1, Exercise 3.2 ======
Solutions of Question 1 of Exercise 3.2 of Unit 03: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.

=====Question.1(i)=====
If $\vec{a}=3\hat{i}-5\hat{j}$ and $\vec{b}=-2\hat{i}+3\hat{j}$, then find $\vec{a}+2\vec{b}$. 	
====Solution====
\begin{align}\vec{a}+2\vec{b}&=3\hat{i}-5\hat{j}+2(-2\hat{i}+3\hat{j})\\
&=3\hat{i}-5\hat{j}-4\hat{i}+6\hat{j}\\
&=-\hat{i}+\hat{j}\end{align}

=====Question.1(ii)=====
If $\vec{a}=3\hat{i}-5\hat{j}$ and $\vec{b}=-2\hat{i}+3\hat{j}$, then find $3\vec{a}-2\vec{b}$. 	
====Solution====
\begin{align}3\vec{a}-2\vec{b}&=3(3\hat{i}-5\hat{j})-2(-2\hat{i}+3\hat{j})\\
&=9\hat{i}-15\hat{j}+4\hat{i}-6\hat{j}\\
&=13\hat{i}-21\hat{j}\end{align}

=====Question.1(iii)=====
If $\vec{a}=3\hat{i}-5\hat{j}$ and $\vec{b}=-2\hat{i}+3\hat{j}$, then find $2(\vec{a}-\vec{b})$. 	
====Solution====
First we have,
\begin{align}\vec{a}-\vec{b}&=3\hat{i}-5\hat{j}-(-2\hat{i}+3\hat{j})\\
&=3\hat{i}-5\hat{j}+2\hat{i}-3\hat{j}\\
&=5\hat{i}-8\hat{j}\end{align}
Multiply both sides by $2$. We have, 
$$2(\vec{a}-\vec{b})=10\hat{i}-16\hat{j}$$

=====Question.1(iv)=====
If $\vec{a}=3\hat{i}-5\hat{j}$ and $\vec{b}=-2\hat{i}+3\hat{j}$, then find $|\vec{a}+\vec{b}|$. 	
====Solution====
We have,
\begin{align}\vec{a}+\vec{b}&=3\hat{i}-5\hat{j}+(-2\hat{i}+3\hat{j})\\
&=3\hat{i}-5\hat{j}-2\hat{i}+3\hat{j}\\
&=\hat{i}-2\hat{j}\end{align}
Taking modulus of both sides. We have,
$$|\vec{a}+\vec{b}|=\sqrt{(1)^2+(-2)^2}=\sqrt{5}$$

=====Question.1(v)=====
If $\vec{a}=3\hat{i}-5\hat{j}$ and $\vec{b}=-2\hat{i}+3\hat{j},$then find $|\vec{a}|-|\vec{b}|.$ 	
====Solution====
First, we find 
\begin{align}|\vec{a}|&=\sqrt(3)^2+(-5)^2=\sqrt{34} …(i)\\
|\vec{b}|&=\sqrt(-2)^2+(3)^2=\sqrt{13} …(ii)\end{align}
Subtracting (i) from (ii). We get
$$|\hat{a}|-|\hat{b}|=\sqrt{34}-\sqrt{13}$$

=====Question.1(vi)=====
If $\vec{a}=3\hat{i}-5\hat{j}$ and $\vec{b}=-2\hat{i}+3\hat{j}$, then find $\dfrac{|\vec{a}|}{|\vec{b}|}$.
$ ====Solution====
Solve yourself.

====Go to ====
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