SATHEE: Vectors 4 Question 15

Vectors 4 Question 15

15. (i) If $C$ is a given non-zero scalar and $\vec{A}$ and $\vec{B}$ be given non-zero vectors such that $\vec{A} ⊥ \vec{B}$ , then find the vector $\vec{X}$ which satisfies the equations $\vec{A} \cdot \vec{X} = c$ and $\vec{A} \times \vec{X} = \vec{B}$

(ii) $\vec{A}$ vector $A$ has components $A_{1}, A_{2}, A_{3}$ in a right-handed rectangular cartesian coordinate system OXYZ. The coordinate system is rotated about the $X$ -axis through an angle $\frac{π}{2}$ . Find the components of $A$ in the new coordinate system, in terms of $A_{1}, A_{2}, A_{3}$ .

(1983, 2M)

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Solution:

(i) Given, $\vec{A} ⊥ \vec{B} \Rightarrow \vec{A} \cdot \vec{B} = 0$

and $\vec{A} \times \vec{X} = \vec{B} \Rightarrow \vec{A} \cdot \vec{B} = 0$ and $\vec{X} \cdot \vec{B} = 0$

Now, $[\vec{X} \vec{A} \vec{A} \times \vec{B}] = \vec{X} \cdot \vec{A} \times (\vec{A} \times \vec{B})$

$= \vec{X} \cdot (\vec{A} \cdot \vec{B}) \vec{A} - (\vec{A} \cdot \vec{A}) \vec{B}$

$= (\vec{A} \cdot \vec{B}) (\vec{X} \cdot \vec{A}) - (\vec{A} \cdot \vec{A}) (\vec{X} \cdot \vec{B}) = 0$

$\Rightarrow \vec{X}, \vec{A}, \vec{A} \times \vec{B}$ are coplanar.

So, $\vec{X}$ can be represented as a linear combination of $\vec{A}$ and $\vec{A} \times \vec{B}$ . Let us consider, $\vec{X} = l \vec{A} + m (\vec{A} \times \vec{B})$

Since,

$\vec{A} \cdot \vec{X} = c$

$∴ \vec{A} \cdot l \vec{A} + m (\vec{A} \times \vec{B}) = c$

$\Rightarrow l | \vec{A} |^{2} + 0 = c$

$\Rightarrow l = \frac{c}{| \vec{A} |^{2}}$

Also, $\vec{A} \times \vec{X} = \vec{B} \Rightarrow \vec{A} \times l \vec{A} + m (\vec{A} \times \vec{B}) = \vec{B}$

$\Rightarrow l (\vec{A} \times \vec{A}) + m \vec{A} \times (\vec{A} \times \vec{B}) = \vec{B}$

$\Rightarrow 0 - m | \vec{A} |^{2} \vec{B} = \vec{B}$

$\Rightarrow m = - \frac{1}{| \vec{A} |^{2}}$

$∴ \vec{X} = (\frac{c}{| \vec{A} |^{2}}) \vec{A} - (\frac{1}{| \vec{A} |^{2}}) (\vec{A} \times \vec{B})$

(ii) Since, vector $\vec{A}$ has components $A_{1}, A_{2}, A_{3}$ in the coordinate system $O X Y Z$ .

$∴ \vec{A} = A_{1} \hat{i} + A_{2} \hat{j} + A_{3} \hat{k}$

When the given system is rotated about an angle of $π / 2$ , the new $X$ -axis is along old $Y$ -axis and new $Y$ -axis is along the old negative $X$ -axis , whereas $z$ remains same.