====== Question 3 & 4, Exercise 3.2 ======
Solutions of Question 3 & 4 of Exercise 3.2 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.

=====Question 3=====
If $\vec{r}=\hat{i}-9\hat{j}$, $\vec{a}=\hat{i}+2\hat{j}$ and $\vec{b}=5\hat{i}-\hat{j}$, determine the real number $p$ and $q$  such that $\vec{r}=p\vec{a}+q\vec{b}$.
====Solution====
Given
$$\vec{r}=p\vec{a}+q\vec{b}.$$
We put the values of $\vec{r},\vec{a}$ and $\vec{b}$ in the given equation. We get
$$\hat{i}-9\hat{j}=p(\hat{i}+2\hat{j})+q(5\hat{i}-\hat{j})$$
$$\implies \hat{i}-9\hat{j}=(p+5q)\hat{i}+(2p-q)\hat{j}.$$
By comparing the coeffients of $\hat{i}$ and $\hat{j}$, we have,
$$p+5q=1…(i)$$
$$2p-q=-9 …(ii)$$
Multiply $2$ by (i) and subtract (ii) from (i). We have
\[\begin{array}{ccc}
2p&+10q&=2 \\  
\mathop+\limits_{-}2p&\mathop-\limits_{+}q&=\mathop-\limits_{+}9 \\ \hline
&11q&=11\\ 
\end{array} \]
$$\implies q=1$$ \\
Put the value of $q$ in (i). We have,
$$p+5(1)=1 \quad \implies p=-4$$
Hence we have $p=-4$ and $q=1$.

=====Question 4=====
If $\vec{p}=2\hat{i}-\hat{j}$ and $\vec{q}=x\hat{i}+3\hat{j},$ then find the value of $x$ such that $|\vec{p}+\vec{q}|=5.$
====Solution==== 
We calculate
\begin{align}\vec{p}+\vec{q}&=2\hat{i}-\hat{j}+x\hat{i}+3\hat{j}\\
&=(2+x)\hat{i}+2\hat{j}\end{align}
Thus
\begin{align}|\vec{p}+\vec{q}|&=\sqrt{(2+x)^2+2^2}\\ 
&=\sqrt{x^2+4x+4+4}\\
&=\sqrt{x^2+4x+8}\end{align}
But we are given that 
\begin{align}&|\vec{p}+\vec{q}|=5 \\
\implies & \sqrt{{{x}^{2}}+4x+8}=5 \\
\implies & x^2+4x+8=25\\
\implies & x^2+4x-17=0\end{align}
This is quadratic equation with $a=1$, $b=4$ and $c=-17$, so
\begin{align}x&=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a} \\
&=\dfrac{-4\pm \sqrt{4^2-4(1)(-17)}}{2(1)}\\
&=\dfrac{-4\pm \sqrt{84}}{2}\\
&=\dfrac{-4\pm 2\sqrt{21}}{2}\\
&=-2\pm \sqrt{21}\end{align}
Thus
$$x=-2\pm\sqrt{21}.$$


====Go To====
<text align="left"><btn type="primary">[[math-11-kpk:sol:unit03:ex3-2-p2 |< Question 2 ]]</btn></text>
<text align="right"><btn type="success">[[math-11-kpk:sol:unit03:ex3-2-p4|Question 5 & 6 >]]</btn></text>