Answer
Verified
394.2k+ views
Hint: We know that the dot product of two perpendicular vectors is equal to zero, using this we’ll form an equation in a, b, c. Then we’ll substitute the values from the option to that equation to check which of the options are correct and hence will acquire the answer.
Complete step by step Answer:
Given data: two vectors $2\hat i + 3\hat j - 4\hat k$ and $a\hat i + b\hat j + c\hat k$
We know that the dot product of two perpendicular vectors is equal to zero
Therefore two vectors $2\hat i + 3\hat j - 4\hat k$ and $a\hat i + b\hat j + c\hat k$ will perpendicular if their dot product is zero
i.e. $(2\hat i + 3\hat j - 4\hat k).(a\hat i + b\hat j + c\hat k) = 0$
using $\hat i.\hat i = \hat j.\hat j{\text{ = }}\hat k.\hat k = 1$, $\hat i.\hat j = \hat j.\hat i = 0$, $\hat i.\hat k = \hat k.\hat i = 0$and $\hat j.\hat k{\text{ = }}\hat k.\hat j = 0$
$ \Rightarrow 2a + 3b - 4c = 0...........(i)$
Now, substituting the values of option in equation(i)
Option(i) \[a = 2,{\text{ }}b = 3,{\text{ }}c = - 4\]
Substituting values in the left-hand side
$ \Rightarrow 2(2) + 3(3) - 4( - 4)$
Since, $2(2) + 3(3) - 4( - 4) \ne 0$
It is not the correct option
Option(ii) \[a = 2,{\text{ }}b = 3,{\text{ }}c = - 4\]
Substituting values in the left-hand side
$ \Rightarrow 2(4) + 3(4) - 4(5)$
$ \Rightarrow 8 + 12 - 20$
$ \Rightarrow 0$
Since, $2(a) + 3(b) - 4(c) = 0$
It is the correct option
Option(B) is correct
Option(iii) \[a = 4,{\text{ }}b = 4,{\text{ }}c = - 5\]
Substituting values in the left-hand side
$ \Rightarrow 2(4) + 3(4) - 4( - 5)$
Since, $2(4) + 3(4) - 4( - 5) \ne 0$
It is not the correct option.
Therefore, from the above results, we can say that option(B) is the correct option
Note: In the above solution we mentioned that the dot product of two perpendicular vectors is zero, the actual formula for the dot product of two vectors is given by
$\vec a.\vec b = \left| {\vec a} \right|\left| {\vec b} \right|\cos \theta $, where $\theta $ is the angle between the two vectors. Now, for perpendicular vectors $\theta = {90^ \circ }$i.e. $\cos \theta = 0$, hence the dot product results in zero.
Complete step by step Answer:
Given data: two vectors $2\hat i + 3\hat j - 4\hat k$ and $a\hat i + b\hat j + c\hat k$
We know that the dot product of two perpendicular vectors is equal to zero
Therefore two vectors $2\hat i + 3\hat j - 4\hat k$ and $a\hat i + b\hat j + c\hat k$ will perpendicular if their dot product is zero
i.e. $(2\hat i + 3\hat j - 4\hat k).(a\hat i + b\hat j + c\hat k) = 0$
using $\hat i.\hat i = \hat j.\hat j{\text{ = }}\hat k.\hat k = 1$, $\hat i.\hat j = \hat j.\hat i = 0$, $\hat i.\hat k = \hat k.\hat i = 0$and $\hat j.\hat k{\text{ = }}\hat k.\hat j = 0$
$ \Rightarrow 2a + 3b - 4c = 0...........(i)$
Now, substituting the values of option in equation(i)
Option(i) \[a = 2,{\text{ }}b = 3,{\text{ }}c = - 4\]
Substituting values in the left-hand side
$ \Rightarrow 2(2) + 3(3) - 4( - 4)$
Since, $2(2) + 3(3) - 4( - 4) \ne 0$
It is not the correct option
Option(ii) \[a = 2,{\text{ }}b = 3,{\text{ }}c = - 4\]
Substituting values in the left-hand side
$ \Rightarrow 2(4) + 3(4) - 4(5)$
$ \Rightarrow 8 + 12 - 20$
$ \Rightarrow 0$
Since, $2(a) + 3(b) - 4(c) = 0$
It is the correct option
Option(B) is correct
Option(iii) \[a = 4,{\text{ }}b = 4,{\text{ }}c = - 5\]
Substituting values in the left-hand side
$ \Rightarrow 2(4) + 3(4) - 4( - 5)$
Since, $2(4) + 3(4) - 4( - 5) \ne 0$
It is not the correct option.
Therefore, from the above results, we can say that option(B) is the correct option
Note: In the above solution we mentioned that the dot product of two perpendicular vectors is zero, the actual formula for the dot product of two vectors is given by
$\vec a.\vec b = \left| {\vec a} \right|\left| {\vec b} \right|\cos \theta $, where $\theta $ is the angle between the two vectors. Now, for perpendicular vectors $\theta = {90^ \circ }$i.e. $\cos \theta = 0$, hence the dot product results in zero.
Recently Updated Pages
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
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Write an application to the principal requesting five class 10 english CBSE
Difference Between Plant Cell and Animal Cell
a Tabulate the differences in the characteristics of class 12 chemistry CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
Discuss what these phrases mean to you A a yellow wood class 9 english CBSE
List some examples of Rabi and Kharif crops class 8 biology CBSE