If \(\rm \vec{a},\vec{b},\vec{c}\) are three non-zero vectors with no two of which are collinear, \(\rm \vec{a}+2\vec{b}\) is collinear with \(\rm \vec {c}\) and \(\rm \vec{b}+3\vec{c}\) is collinear with \(\rm \vec {a}\), then \(\rm |\vec{a} +2 \vec{b}+6\vec{c}|\) will be equal to

Cocept:

If a vector A is collinear to other vector B, it can be written as:

\(\rm \vec A = λ × \vec B\) where λ is any scalar number

Calculation:

Given \(\rm \vec{a}+2\vec{b}\) is collinear with \(\rm \vec {c}\) 

⇒ \(\rm \vec{a}+2\vec{b}\) = λ₁ × \(\rm \vec {c}\)                      ...(i)

Also given \(\rm \vec{b}+3\vec{c}\) is collinear with \(\rm \vec {a}\)

⇒ \(\rm \vec{b}+3\vec{c}\) = λ₂ × \(\rm \vec {a}\)                     ...(ii)

Subtracting 2×(ii) from (i)

⇒ \(\rm \vec{a}-6\vec{c} = λ_1 \vec c-2λ_2 \vec a\)

∴ λ₁ = - 6, λ₂ = \(\rm {-1\over2}\) 

From (ii)

⇒ \(\rm \vec{b}+3\vec{c}\)= \(\rm {-1\over2}\)\(\rm \vec {a}\)

⇒ \(\rm 2\vec{b}+6\vec{c}\)= \(\rm -\vec {a}\)

⇒ \(\rm \vec a+ 2\vec{b}+6\vec{c} = 0\)

⇒ \(\boldsymbol{\rm \left|\vec a+ 2\vec{b}+6\vec{c}\right| = 0}\)