Question

35.
Three vectors A = ai +j+k; B=i+ bj +
and C = i + i + ck are mutually perpendicular
(i. j and ħ are unit vectors along X, Y and
Z axis respectively). The respective values of
a, b and care
[WB JEE 2017]​

Answer 1

Proper Question: $\boldsymbol{Three\:vectors:\vec{A}=a\hat{\textbf{\i}}+\hat{\textbf{\j}}+\hat{\textbf{k}},\:\vec{B}=\hat{\textbf{\i}}+b\hat{\textbf{\j}}+\hat{\textbf{k}}\:\:and\:\:\vec{C}=\hat{\textbf{\i}}+\hat{\textbf{\j}}+c\hat{\textbf{k}}}$

are mutually perpendicular to each other. (where $\hat{\textbf{\i}}$. $\hat{\textbf{\j}}$ and $\hat{\textbf{k}}$ are unit vectors along X, Y and Z axis respectively). The respective values of a, b and c are?

Answer: The values of a, b and c are equal to -1/2, -1/2 and -1/2 respectively.

Given:

$\boldsymbol{The\:three\:vectors\:are:\vec{A}=a\hat{\textbf{\i}}+\hat{\textbf{\j}}+\hat{\textbf{k}},\:\vec{B}=\hat{\textbf{\i}}+b\hat{\textbf{\j}}+\hat{\textbf{k}}\:\:and\:\:\vec{C}=\hat{\textbf{\i}}+\hat{\textbf{\j}}+c\hat{\textbf{k}}}$The three vectors are mutually perpendicular to each other.

To Find:

The values of a, b and c.

Solution:

→ The dot product or scalar product of two vectors $\boldsymbol{\vec{P}}$ and $\boldsymbol{\vec{Q}}$ is given as:

   $\boldsymbol{\vec{P}=(p_{1}) \hat{\textbf{\i}}+(p_{2}) \hat{\textbf{\j}}+(p_{3} )\hat{\textbf{k}}}$

   $\boldsymbol{\vec{Q}=(q_{1}) \hat{\textbf{\i}}+(q_{2}) \hat{\textbf{\j}}+(q_{3} )\hat{\textbf{k}}}$

$\boldsymbol{\therefore Dot\:Product=\vec{P}.\vec{Q}=|\vec{P}||\vec{Q}|cos\theta=(p_{1}q_{1} +p_{2} q_{2} +p_{3} q_{3} )}$

   where 'θ' is the angle between the two vectors $\boldsymbol{\vec{P}}$ and $\boldsymbol{\vec{Q}}$.

→ Some important properties of the dot product or scalar product are:

(i) The value of the dot product is maximum if the angle between the two vectors is equal to zero that is if the two vectors are parallel.

(ii) The value of the dot product is zero if the angle between the two vectors is equal to 90° that is if the two vectors are perpendicular.

→ In the given question:

$\boldsymbol{The\:three\:vectors\:are:\vec{A}=a\hat{\textbf{\i}}+\hat{\textbf{\j}}+\hat{\textbf{k}},\:\vec{B}=\hat{\textbf{\i}}+b\hat{\textbf{\j}}+\hat{\textbf{k}}\:\:and\:\:\vec{C}=\hat{\textbf{\i}}+\hat{\textbf{\j}}+c\hat{\textbf{k}}}$The three vectors are mutually perpendicular to each other.

→ As the three vectors are perpendicular to one another therefore their dot product or scalar product must be equal to zero.

(i) The vectors $\boldsymbol{\vec{A}}$ and $\boldsymbol{\vec{B}}$ are perpendicular to each other:

                           $\boldsymbol{\therefore \vec{A}.\vec{B}=(a+b+1)=0}\\\\\boldsymbol{\therefore (a+b)=(-1)}\\\\\boldsymbol{\therefore a=-(b+1)-Equation(i)}$

(i) The vectors $\boldsymbol{\vec{B}}$ and $\boldsymbol{\vec{C}}$ are perpendicular to each other:

                           $\boldsymbol{\therefore \vec{B}.\vec{C}=(1+b+c)=0}\\\\\boldsymbol{\therefore (b+c)=(-1)}\\\\\boldsymbol{\therefore c=-(b+1)-Equation(ii)}$

(i) The vectors $\boldsymbol{\vec{A}}$ and $\boldsymbol{\vec{C}}$ are perpendicular to each other:

                           $\boldsymbol{\therefore \vec{A}.\vec{C}=(a+1+c)=0}\\\\\boldsymbol{\therefore (a+c)=(-1)-Equation(iii)}$

Putting the values of 'a' and 'c' from Equations (i) and (ii) into Equation (iii):

                                   ∵ a + c = -1

                                   ∴ -(b + 1) - (b + 1) = -1

                                   ∴ -2b - 2 = -1

                                   ∴ -2b = 1

                                   ∴ b = -1/2

→ The value of 'b' comes out to be equal to -1/2.

→ Putting the value of 'b' into Equation (i):

                                   ∵ a = -(b + 1)

                                   ∴ a = -(-1/2) - 1

                                   ∴ a = 1/2 - 1

                                   ∴ a = -1/2

→ The value of 'a' comes out to be equal to -1/2.

→ Putting the value of 'b' into Equation (ii):

                                    ∵ c = -(b + 1)

                                    ∴ c = -(-1/2) - 1

                                    ∴ c = 1/2 - 1

                                    ∴ c = -1/2

→ The value of 'c' comes out to be equal to -1/2.

Therefore the values of a, b and c come out to be equal to -1/2, -1/2 and -1/2 respectively.

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