Question

If $a$ is a unit vector satisfying $a\times r=b$, $a\cdot r=c$ and $a\cdot b=0$. Then
determine the value of $r$ .
(a) $cb+\left( a\times b \right)$
(b) $ca+\left( a\times b \right)$
(c) $cb-\left( a\times b \right)$
(d) $ca-\left( a\times b \right)$

Answer 1

In this question, we will first evaluate the value of $a\times \left( a\times r \right)$ given that $a\times r=b$. Then using the triple cross product formula $\overrightarrow{a}\times \left( \overrightarrow{b}\times \overrightarrow{c} \right)=\left( \overrightarrow{a}\cdot \overrightarrow{c} \right)\overrightarrow{b}-\left( \overrightarrow{a}\cdot \overrightarrow{b} \right)\overrightarrow{c}$ for vectors $\overrightarrow{a}$, $\overrightarrow{b}$ and
$\overrightarrow{c}$, we will then find the value of $a\times \left( a\times r \right)$. Then using the fact that $a$ is a unit vector we have ${{a}^{2}}=1$. We will then substitute the value ${{a}^{2}}=1$ and $a\cdot r=c$ in the expression for $a\times \left( a\times r \right)$ to get the value of $r$.

Complete step-by-step answer:
We are given a unit vector $a$.
$\Rightarrow {{a}^{2}}=1$
Since we have $a\times r=b$, thus on evaluating the value of $a\times \left( a\times r \right)$ by
substituting $a\times r=b$ we get
$a\times \left( a\times r \right)=a\times b$
Now we know that for vectors $\overrightarrow{a}$, $\overrightarrow{b}$ and
$\overrightarrow{c}$, then the triple cross product of vectors $\overrightarrow{a}$,
$\overrightarrow{b}$ and $\overrightarrow{c}$is given by

\overrightarrow{a}\cdot \overrightarrow{c} \right)\overrightarrow{b}-\left( \overrightarrow{a}\cdot \overrightarrow{b} \right)\overrightarrow{c}$$ Using the above formula for the triple cross product in $$a\times \left( a\times r \right)$$, we get that $$a\times \left( a\times r \right)=\left( a\cdot r \right)a-\left( a\cdot a \right)r$$ Now, using $$a\cdot r=c$$ in the above equation we have $$\begin{aligned} & a\times \left( a\times r \right)=\left( a\cdot r \right)a-\left( a\cdot a \right)r \\\ & =ca-{{a}^{2}}r \end{aligned}$$ Since $$a\times \left( a\times r \right)=a\times b$$ , thus we have $$ca-{{a}^{2}}r=a\times b...........(1)$$ Also $${{a}^{2}}=1$$where $$a$$is a unit vector. Therefore substituting the value $${{a}^{2}}=1$$ in equation (1) , we get $$ca-r=a\times b$$ We will now calculate the value of $$r$$ by rearranging the terms of the above equation. By taking vector $$r$$ to the right and taking $$a\times b$$ to the left side of the equation we get $$ca-\left( a\times b \right)=r$$ Therefore the value of $$r$$ is given by $$ca-\left( a\times b \right)$$ **So, the correct answer is “Option (d)”.** **Note:** In this problem, we have also use the definition of cross product and dot product to get the desired value of $$r$$. We have $$a\times b=ab\sin \theta $$ where $$\theta $$ is the angle between $$a$$ and $$b$$. Also $$a\cdot b=ab\cos \theta $$. Then using the fact that $$a\cdot b=0$$, we say that vectors $$a$$ and $$b$$ are orthogonal to each other. That is the vectors $$a$$ and $$b$$ are perpendicular to each other.