Question

any three vectors such that $\vec{a}\cdot\vec{b} \ne 0, \vec{b}\cdot\vec{c} \ne 0,$ then $\vec{a}$ and $\vec{c}$ are :

• A. inclined at an angle of $\frac{\pi}{6}$ between them
• B. perpendicular
• C. parallel
• D. inclined at an angle of $\frac{\pi}{3}$ between them

Answer 1

Answer: (C)

parallel

Answer 2

$\because \left(\vec{a}\times\vec{b}\right)\times\vec{c}=\vec{a}\times\left(\vec{b}\times\vec{c}\right)$ $\Rightarrow \left(\vec{a}\cdot\vec{c}\right)\vec{b}-\left(\vec{b}\cdot\vec{c}\right)\vec{a}=\left(\vec{a}\cdot\vec{c}\right)\vec{b}-\left(\vec{a}\cdot\vec{b}\right)\vec{c}$ $\Rightarrow \left(\vec{b}\cdot\vec{c}\right) \vec{a}=\left(\vec{a}\cdot\vec{b}\right)\vec{c}$ $\Rightarrow \vec{a}$ is parallel to $\vec{c}.$