Answer
Verified
394.2k+ views
Hint: First, before proceeding for this, we must know the concept that in vector addition and subtraction only like components get added or subtracted to get the value of vector$\overrightarrow{b}+\overrightarrow{c}$. Then, let us suppose that n be the unit vector along $\overrightarrow{b}+\overrightarrow{c}$ and the value of the unit vector n is given by $\widehat{n}=\dfrac{\overrightarrow{b}+\overrightarrow{c}}{\left| \overrightarrow{b}+\overrightarrow{c} \right|}$. Then, in the question, we are given the condition that the above unit vector n and the vector a as scalar product as 1 to get the value of $\lambda $. Then, by substituting the value of $\lambda $ in the unit n vector to get its value.
Complete step-by-step answer:
In this question, we are supposed to find the value of $\lambda $from the condition given that the scalar product of the vector $\overrightarrow{a}=\widehat{i}+\widehat{j}+\widehat{k}$ with a unit vector along the sum of vectors $\overrightarrow{b}=2\widehat{i}+4\widehat{j}-5\widehat{k}$ and $\overrightarrow{c}=\lambda \widehat{i}+2\widehat{j}+3\widehat{k}$ is equal to one and also find the unit vector along $\overrightarrow{b}+\overrightarrow{c}$.
So, before proceeding for this, we must know the concept that in vector addition and subtraction only like components get added or subtracted.
Then, by using the above concept, we need to find the value of the summation of vectors of $\overrightarrow{b}+\overrightarrow{c}$as:
$\overrightarrow{b}+\overrightarrow{c}=2\widehat{i}+4\widehat{j}-5\widehat{k}+\lambda \widehat{i}+2\widehat{j}+3\widehat{k}$
So, by solving the above expression, we get:
$\overrightarrow{b}+\overrightarrow{c}=\left( 2+\lambda \right)\widehat{i}+6\widehat{j}-2\widehat{k}$
Now, let us suppose that n be the unit vector along $\overrightarrow{b}+\overrightarrow{c}$ and the value of the unit vector n is given by:
$\widehat{n}=\dfrac{\overrightarrow{b}+\overrightarrow{c}}{\left| \overrightarrow{b}+\overrightarrow{c} \right|}$
Then, by solving for the unit vector n, we get:
$\begin{align}
& \widehat{n}=\dfrac{\left( 2+\lambda \right)\widehat{i}+6\widehat{j}-2\widehat{k}}{\sqrt{{{\left( 2+\lambda \right)}^{2}}+{{6}^{2}}+{{\left( -2 \right)}^{2}}}} \\
& \Rightarrow \widehat{n}=\dfrac{\left( 2+\lambda \right)\widehat{i}+6\widehat{j}-2\widehat{k}}{\sqrt{{{\lambda }^{2}}+4+4\lambda +36+4}} \\
& \Rightarrow \widehat{n}=\dfrac{\left( 2+\lambda \right)\widehat{i}+6\widehat{j}-2\widehat{k}}{\sqrt{{{\lambda }^{2}}+4\lambda +44}} \\
\end{align}$
Now, in the question, we are given with the condition that the above unit vector n and the vector a as scalar product as 1.
So, by using this condition, we get:
$\overrightarrow{a}\centerdot \widehat{n}=1$
Now, by substituting the value of vector a and unit vector n calculated above, we get:
$\left( \widehat{i}+\widehat{j}+\widehat{k} \right)\centerdot \left( \dfrac{\left( 2+\lambda \right)\widehat{i}+6\widehat{j}-2\widehat{k}}{\sqrt{{{\lambda }^{2}}+4\lambda +44}} \right)=1$
Then, by solving the above expression, we get:
$\begin{align}
& \left( 2+\lambda \right)+6-2=\sqrt{{{\lambda }^{2}}+4\lambda +44} \\
& \Rightarrow \lambda +6=\sqrt{{{\lambda }^{2}}+4\lambda +44} \\
\end{align}$
Then, by squaring both sides, we get:
$\begin{align}
& {{\left( \lambda +6 \right)}^{2}}={{\lambda }^{2}}+4\lambda +44 \\
& \Rightarrow {{\lambda }^{2}}+12\lambda +36={{\lambda }^{2}}+4\lambda +44 \\
& \Rightarrow 8\lambda =8 \\
& \Rightarrow \lambda =1 \\
\end{align}$
So, we get the value of $\lambda $ as 1.
Now, by substituting the value of $\lambda $ as 1 in unit n vector to get its value as:
$\begin{align}
& \widehat{n}=\dfrac{\left( 2+1 \right)\widehat{i}+6\widehat{j}-2\widehat{k}}{\sqrt{{{1}^{2}}+4\left( 1 \right)+44}} \\
& \Rightarrow \widehat{n}=\dfrac{3\widehat{i}+6\widehat{j}-2\widehat{k}}{\sqrt{1+4+44}} \\
& \Rightarrow \widehat{n}=\dfrac{3\widehat{i}+6\widehat{j}-2\widehat{k}}{\sqrt{49}} \\
& \Rightarrow \widehat{n}=\dfrac{3\widehat{i}+6\widehat{j}-2\widehat{k}}{7} \\
\end{align}$
So, we get the value of unit vector along $\overrightarrow{b}+\overrightarrow{c}$ as $\widehat{n}=\dfrac{3\widehat{i}+6\widehat{j}-2\widehat{k}}{7}$.
Note: To solve these types of questions we need to know some of the basic calculations of the dot product or scalar product. So, if we have two vectors as $X\left( a\widehat{i}+b\widehat{j}+c\widehat{k} \right)$ and $Y\left( d\widehat{i}+e\widehat{j}+f\widehat{k} \right)$, then their scalar product is given by:
$X\centerdot Y=a\times d+b\times e+c\times f$.
Complete step-by-step answer:
In this question, we are supposed to find the value of $\lambda $from the condition given that the scalar product of the vector $\overrightarrow{a}=\widehat{i}+\widehat{j}+\widehat{k}$ with a unit vector along the sum of vectors $\overrightarrow{b}=2\widehat{i}+4\widehat{j}-5\widehat{k}$ and $\overrightarrow{c}=\lambda \widehat{i}+2\widehat{j}+3\widehat{k}$ is equal to one and also find the unit vector along $\overrightarrow{b}+\overrightarrow{c}$.
So, before proceeding for this, we must know the concept that in vector addition and subtraction only like components get added or subtracted.
Then, by using the above concept, we need to find the value of the summation of vectors of $\overrightarrow{b}+\overrightarrow{c}$as:
$\overrightarrow{b}+\overrightarrow{c}=2\widehat{i}+4\widehat{j}-5\widehat{k}+\lambda \widehat{i}+2\widehat{j}+3\widehat{k}$
So, by solving the above expression, we get:
$\overrightarrow{b}+\overrightarrow{c}=\left( 2+\lambda \right)\widehat{i}+6\widehat{j}-2\widehat{k}$
Now, let us suppose that n be the unit vector along $\overrightarrow{b}+\overrightarrow{c}$ and the value of the unit vector n is given by:
$\widehat{n}=\dfrac{\overrightarrow{b}+\overrightarrow{c}}{\left| \overrightarrow{b}+\overrightarrow{c} \right|}$
Then, by solving for the unit vector n, we get:
$\begin{align}
& \widehat{n}=\dfrac{\left( 2+\lambda \right)\widehat{i}+6\widehat{j}-2\widehat{k}}{\sqrt{{{\left( 2+\lambda \right)}^{2}}+{{6}^{2}}+{{\left( -2 \right)}^{2}}}} \\
& \Rightarrow \widehat{n}=\dfrac{\left( 2+\lambda \right)\widehat{i}+6\widehat{j}-2\widehat{k}}{\sqrt{{{\lambda }^{2}}+4+4\lambda +36+4}} \\
& \Rightarrow \widehat{n}=\dfrac{\left( 2+\lambda \right)\widehat{i}+6\widehat{j}-2\widehat{k}}{\sqrt{{{\lambda }^{2}}+4\lambda +44}} \\
\end{align}$
Now, in the question, we are given with the condition that the above unit vector n and the vector a as scalar product as 1.
So, by using this condition, we get:
$\overrightarrow{a}\centerdot \widehat{n}=1$
Now, by substituting the value of vector a and unit vector n calculated above, we get:
$\left( \widehat{i}+\widehat{j}+\widehat{k} \right)\centerdot \left( \dfrac{\left( 2+\lambda \right)\widehat{i}+6\widehat{j}-2\widehat{k}}{\sqrt{{{\lambda }^{2}}+4\lambda +44}} \right)=1$
Then, by solving the above expression, we get:
$\begin{align}
& \left( 2+\lambda \right)+6-2=\sqrt{{{\lambda }^{2}}+4\lambda +44} \\
& \Rightarrow \lambda +6=\sqrt{{{\lambda }^{2}}+4\lambda +44} \\
\end{align}$
Then, by squaring both sides, we get:
$\begin{align}
& {{\left( \lambda +6 \right)}^{2}}={{\lambda }^{2}}+4\lambda +44 \\
& \Rightarrow {{\lambda }^{2}}+12\lambda +36={{\lambda }^{2}}+4\lambda +44 \\
& \Rightarrow 8\lambda =8 \\
& \Rightarrow \lambda =1 \\
\end{align}$
So, we get the value of $\lambda $ as 1.
Now, by substituting the value of $\lambda $ as 1 in unit n vector to get its value as:
$\begin{align}
& \widehat{n}=\dfrac{\left( 2+1 \right)\widehat{i}+6\widehat{j}-2\widehat{k}}{\sqrt{{{1}^{2}}+4\left( 1 \right)+44}} \\
& \Rightarrow \widehat{n}=\dfrac{3\widehat{i}+6\widehat{j}-2\widehat{k}}{\sqrt{1+4+44}} \\
& \Rightarrow \widehat{n}=\dfrac{3\widehat{i}+6\widehat{j}-2\widehat{k}}{\sqrt{49}} \\
& \Rightarrow \widehat{n}=\dfrac{3\widehat{i}+6\widehat{j}-2\widehat{k}}{7} \\
\end{align}$
So, we get the value of unit vector along $\overrightarrow{b}+\overrightarrow{c}$ as $\widehat{n}=\dfrac{3\widehat{i}+6\widehat{j}-2\widehat{k}}{7}$.
Note: To solve these types of questions we need to know some of the basic calculations of the dot product or scalar product. So, if we have two vectors as $X\left( a\widehat{i}+b\widehat{j}+c\widehat{k} \right)$ and $Y\left( d\widehat{i}+e\widehat{j}+f\widehat{k} \right)$, then their scalar product is given by:
$X\centerdot Y=a\times d+b\times e+c\times f$.
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is the stopping potential when the metal with class 12 physics JEE_Main
The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main
Using the following information to help you answer class 12 chemistry CBSE
Trending doubts
State the differences between manure and fertilize class 8 biology CBSE
Why are xylem and phloem called complex tissues aBoth class 11 biology CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
What would happen if plasma membrane ruptures or breaks class 11 biology CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
What precautions do you take while observing the nucleus class 11 biology CBSE
What would happen to the life of a cell if there was class 11 biology CBSE
Change the following sentences into negative and interrogative class 10 english CBSE