Answer

Verified

339.9k+ views

**Hint:**Since the vector $\vec{a}$ is given to be perpendicular to both of the unit vectors $\vec{b}$ and $\vec{c}$, so it will also be perpendicular to the plane containing the vectors $\vec{b}$ and $\vec{c}$. This means that the vector $\vec{a}$ must be parallel to the vector $\vec{b}\times \vec{c}$ . So the vector $\vec{b}\times \vec{c}$ can be written as a scalar multiple of the vector $\vec{a}$. Further, the magnitude of the cross product $\vec{b}\times \vec{c}$ is given by $\left| \vec{b}\times \vec{c} \right|=\left| {\vec{b}} \right|\left| {\vec{c}} \right|\sin \alpha $. Equating this magnitude with the scalar multiplied by the magnitude of $\vec{a}$, we will get the value of the scalar.

**Complete step-by-step solution:**

According to the question, the vector $\vec{a}$ is perpendicular to the unit vectors $\vec{b}$ and $\vec{c}$. This means that it will also be perpendicular to the plane containing the vectors $\vec{b}$ and $\vec{c}$. Now, we know that the cross product of two vectors is perpendicular to the plane containing the two vectors. So we can say that the vector $\vec{a}$ is parallel to the vector $\vec{b}\times \vec{c}$. So we can write

$\Rightarrow \vec{b}\times \vec{c}=k\vec{a}.......(i)$

Taking the magnitudes of the vectors on both the sides, we get

\[\Rightarrow \left| \vec{b}\times \vec{c} \right|=\left| k\vec{a} \right|\]

Now, since the angle between the vectors $\vec{b}$ and $\vec{c}$ is given to be equal to $\alpha $, so we can write

$\Rightarrow \left| \vec{b}\times \vec{c} \right|=\left| {\vec{b}} \right|\left| {\vec{c}} \right|\sin \alpha $

Equating the above two equations, we get

\[\begin{align}

& \Rightarrow \left| k\vec{a} \right|=\left| {\vec{b}} \right|\left| {\vec{c}} \right|\sin \alpha \\

& \Rightarrow k\left| {\vec{a}} \right|=\left| {\vec{b}} \right|\left| {\vec{c}} \right|\sin \alpha \\

\end{align}\]

Since the vectors $\vec{b}$ and $\vec{c}$ are given to be unit vectors, so their magnitudes are equal to one. So we substitute $\left| {\vec{b}} \right|=1$ and $\left| {\vec{c}} \right|=1$ in the above equation to get

\[\begin{align}

& \Rightarrow \left| k \right|\left| {\vec{a}} \right|=\left( 1 \right)\left( 1 \right)\sin \alpha \\

& \Rightarrow \left| k \right|\left| {\vec{a}} \right|=\sin \alpha \\

& \Rightarrow \left| k \right|=\sin \alpha \\

& \Rightarrow k=\pm \sin \alpha \\

\end{align}\]

Substituting this value in (i) we get

$\Rightarrow \vec{b}\times \vec{c}=\pm \left( \sin \alpha \right)\vec{a}$

**Hence, the correct answer is option (c).**

**Note:**In this question, we were not given any information regarding the magnitude of the vector $\vec{a}$. So we simply assumed its magnitude to be equal to one. In other words, we assumed the vector $\vec{a}$ as a unit vector.

Recently Updated Pages

Basicity of sulphurous acid and sulphuric acid are

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

Why should electric field lines never cross each other class 12 physics CBSE

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Explain why a There is no atmosphere on the moon b class 11 physics CBSE

Difference Between Plant Cell and Animal Cell

Can anyone list 10 advantages and disadvantages of friction

State and prove Bernoullis theorem class 11 physics CBSE

The ice floats on water because A solid have lesser class 9 chemistry CBSE

State Newtons formula for the velocity of sound in class 11 physics CBSE