First slide

Algebra of vectors

Question

$Let \vec{a}, \vec{b} and \vec{c} be three non-zero vectors, no two of which are collinear. If the$ $vector \vec{a} + 2 \vec{b} is collinear with \vec{c}, and \vec{b} + 3 \vec{c} is collinear with, \vec{a} then$ $\vec{a} + 2 \vec{b} + 6 \vec{c} is equal to$

Moderate

A

$λ a$
B

$λ b$
C

$λ c$
D

0

Solution

$Let \vec{a} + 2 \vec{b} = x \vec{c} and \vec{b} + 3 \vec{c} = y \vec{a}, where x,y are scalars Then \vec{a} + 2 \vec{b} + 6 \vec{c} = (x + 6) \vec{c} and also 2(\vec{b} +3 \vec{c})=2y \vec{a}$
$\Rightarrow \vec{a} + 2 \vec{b} + 6 \vec{c} = (1 + 2 y) \vec{a} S (x + 6) \vec{c} = (1 + 2 y) \vec{a} . Since \vec{a} and \vec{c} are non-zero and non-collinear,$
$we have x + 6 = 0 and 1 + 2 y = 0, i.e., x = - 6 and y = - 1 / 2 . In either case, we have$ $\vec{a} + 2 \vec{b} + 6 \vec{c} = 0 .$

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