====== Question 7 Exercise 3.5 ======
Solutions of Question 7 of Exercise 3.5 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 7(i)=====
For what value of $c$ the following vectors are coplanar $\vec{u}=\hat{i}+2 \hat{j}+3 \hat{k}$.
$\vec{v}=2 \hat{i}-3 \hat{j}+4 \hat{k} \cdot \vec{w}=3 \hat{i}+\hat{j}+c \hat{k}$
====Solution====
The given vectors are coplanar, therefore
\begin{align}\vec{u} \cdot \vec{v} \times \vec{w}&=0\\
\vec{u} \cdot \vec{v} \times \vec{w}&=0\\
\Rightarrow\left|\begin{array}{ccc}1 & 2 & 3 \\ 2 & -3 & 4 \\ 3 & 1 & c\end{array}\right|&=0\\
1(-3 c-4)-2(2 c-12)+3(2+9)&=0\\
\Rightarrow-3 c-4-4 c+24+33&=0\\
\Rightarrow \quad-7 c+53&=0\\
\Rightarrow c&=\dfrac{53}{7}.\end{align}
which is required value of $c$ for which the given vectors become coplanar.

=====Question 7(ii)=====
For what value of $c$ the following vectors are coplanar $\vec{u}=\hat{i}+\hat{j}-\hat{k}$.
$\vec{v}=\hat{i}-2 \hat{j}+\hat{k}, \vec{w}=c \hat{i}+\hat{j}-c \hat{k}$.
====Solution====
The given vectors are coplanar, therefore
\begin{align}\vec{u} \cdot \vec{v} \times \vec{w}&=0\\
\Rightarrow\left|\begin{array}{ccc}
1 & 1 & -1 \\
1 & -2 & 1 \\
c & 1 & -c
\end{array}\right|&=0\\
(2 c-1)-(-c-c)-(1+2 c)&=0 \\
\Rightarrow 2 c-1+2 c-1-2 c&=0 \\
\Rightarrow 2 c-2&=0\\
\Rightarrow c&=1\end{align}
which is the required value of $c$ for which the given vectors become coplanar.

=====Question 7(iii)=====
For what value of $c$ the following vectors are coplanar $\vec{u}=\hat{i}+\hat{j}+2 \hat{k}, \vec{v}=2 \hat{i}+3 \hat{j}+\hat{k}$. $\vec{n}=c \hat{i}+2 \hat{j}+6 \hat{k}$
====Solution====
Since the given vectors are coplanar, therefore
\begin{align}\vec{u} \cdot \bar{v} \times \vec{w}&=0 \\
\Rightarrow\left|\begin{array}{lll}
1 & 1 & 2 \\
2 & 3 & 1 \\
c & 2 & 6
\end{array}\right|&=0\\
1(18-2)-1(12-c)+2(4-3 c)&=0 \\
16-12+c+8-6 c&=0 \\
\Rightarrow -5 c+12&=0 \\
\Rightarrow c&=\dfrac{-12}{-5}=\dfrac{12}{5}\end{align}
which is the required value of $c$ for which the given vectors become coplanar.





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