Answer
Verified
394.2k+ views
Hint: To solve this problem we need to know that when two points $\left( {{a}_{1}},{{b}_{1}},{{c}_{1}} \right)\,and\,\left( {{a}_{2}},{{b}_{2}},{{c}_{2}} \right)$ in 3D are given then the equation of the line passing through both of them is given by $\dfrac{x-{{a}_{1}}}{{{a}_{1}}-{{a}_{2}}}=\dfrac{y-{{b}_{1}}}{{{b}_{2}}-{{b}_{1}}}=\dfrac{z-{{c}_{1}}}{{{c}_{1}}-{{c}_{2}}}$ after finding the equation of the line we will equate the equation of the line with some variable say k, and then represent any point on the line in its general form and will put that point in the equation of the given plant to find the value of k.
Complete step by step answer:
We know that when two points $\left( {{a}_{1}},{{b}_{1}},{{c}_{1}} \right)\,and\,\left( {{a}_{2}},{{b}_{2}},{{c}_{2}} \right)$ in 3D are given then the equation of the line passing through both of them is given by $\dfrac{x-{{a}_{1}}}{{{a}_{1}}-{{a}_{2}}}=\dfrac{y-{{b}_{1}}}{{{b}_{1}}-{{b}_{2}}}=\dfrac{z-{{c}_{1}}}{{{c}_{1}}-{{c}_{2}}}$,
Points that are given to us are point (3, -4, -5) and (2, -3, 1) so equation of the plane we get as,
$\begin{align}
& \dfrac{x-3}{3-2}=\dfrac{y+4}{-4+3}=\dfrac{z+5}{-5-1} \\
& \dfrac{x-3}{1}=\dfrac{y+4}{-1}=\dfrac{z+5}{-6} \\
\end{align}$
Now we will find any general point on the above line and for that we will equate the equation of the above lint to a random variable let’s say k,
$\dfrac{x-3}{1}=\dfrac{y+4}{-1}=\dfrac{z+5}{-6}=k$
And any general point we get as,
$\left( k+3,-4-k,-5-6k \right)$
Now if we put the above coordinate in the given plane to find the value of k we will find the coordinates of the intersection point of the line and the plane.
Equation of the plane is 2x + y + z = 7.
Now, Putting$\left( x,y,z \right)=\left( k+3,-4-k,-5-6k \right)$, we get
$\begin{align}
& 2\left( k+3 \right)+\left( -4-k \right)+\left( -5-6k \right)=7 \\
& 2k+6-4-k-5-6k=7 \\
& -5k-3=7 \\
& k=-\dfrac{10}{5} \\
& k=-2 \\
\end{align}$
Hence point which passes through the line made by the points (3, -4, -5) and (2, -3, 1) as well as crosses the plane 2x + y + z = 7 is,
$\begin{align}
& \left( k+3,-4-k,-5-6k \right) \\
& \left( -2+3,-4+2,-5+12 \right)(k=-2) \\
& \left( 1,-2,7 \right) \\
\end{align}$
Note:
While solving the questions of 3D vectors we also need to imagine the problem given in 3D, from that we can easily understand what steps we have to follow in order to solve the question.
In the above problem we could also have written the equation of the line as $\dfrac{x-{{a}_{1}}}{{{a}_{2}}-{{a}_{1}}}=\dfrac{y-{{b}_{1}}}{{{b}_{2}}-{{b}_{1}}}=\dfrac{z-{{c}_{1}}}{{{c}_{2}}-{{c}_{1}}}$ in this expression denominator part of each term is reversed but it really does not matter as it is changed for the every term so only general expression will change but we will get the same answer.
Complete step by step answer:
We know that when two points $\left( {{a}_{1}},{{b}_{1}},{{c}_{1}} \right)\,and\,\left( {{a}_{2}},{{b}_{2}},{{c}_{2}} \right)$ in 3D are given then the equation of the line passing through both of them is given by $\dfrac{x-{{a}_{1}}}{{{a}_{1}}-{{a}_{2}}}=\dfrac{y-{{b}_{1}}}{{{b}_{1}}-{{b}_{2}}}=\dfrac{z-{{c}_{1}}}{{{c}_{1}}-{{c}_{2}}}$,
Points that are given to us are point (3, -4, -5) and (2, -3, 1) so equation of the plane we get as,
$\begin{align}
& \dfrac{x-3}{3-2}=\dfrac{y+4}{-4+3}=\dfrac{z+5}{-5-1} \\
& \dfrac{x-3}{1}=\dfrac{y+4}{-1}=\dfrac{z+5}{-6} \\
\end{align}$
Now we will find any general point on the above line and for that we will equate the equation of the above lint to a random variable let’s say k,
$\dfrac{x-3}{1}=\dfrac{y+4}{-1}=\dfrac{z+5}{-6}=k$
And any general point we get as,
$\left( k+3,-4-k,-5-6k \right)$
Now if we put the above coordinate in the given plane to find the value of k we will find the coordinates of the intersection point of the line and the plane.
Equation of the plane is 2x + y + z = 7.
Now, Putting$\left( x,y,z \right)=\left( k+3,-4-k,-5-6k \right)$, we get
$\begin{align}
& 2\left( k+3 \right)+\left( -4-k \right)+\left( -5-6k \right)=7 \\
& 2k+6-4-k-5-6k=7 \\
& -5k-3=7 \\
& k=-\dfrac{10}{5} \\
& k=-2 \\
\end{align}$
Hence point which passes through the line made by the points (3, -4, -5) and (2, -3, 1) as well as crosses the plane 2x + y + z = 7 is,
$\begin{align}
& \left( k+3,-4-k,-5-6k \right) \\
& \left( -2+3,-4+2,-5+12 \right)(k=-2) \\
& \left( 1,-2,7 \right) \\
\end{align}$
Note:
While solving the questions of 3D vectors we also need to imagine the problem given in 3D, from that we can easily understand what steps we have to follow in order to solve the question.
In the above problem we could also have written the equation of the line as $\dfrac{x-{{a}_{1}}}{{{a}_{2}}-{{a}_{1}}}=\dfrac{y-{{b}_{1}}}{{{b}_{2}}-{{b}_{1}}}=\dfrac{z-{{c}_{1}}}{{{c}_{2}}-{{c}_{1}}}$ in this expression denominator part of each term is reversed but it really does not matter as it is changed for the every term so only general expression will change but we will get the same answer.
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