Let a, b and c be three non-zero vectors such that b and c are non-collinear.

Question

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three non-zero vectors such that \(\vec{b}\) and \(\vec{c}\) are non-collinear. If \(\vec{a}+5 \vec{b}\) is collinear with \(\vec{c}, \vec{b}+6 \vec{c}\) is collinear with \(\vec{a}\) and \(\vec{a}+\alpha \vec{b}+\beta \vec{c}=\vec{0}\), then \(\alpha+\beta\) is equal to

(1) 35

(2) 30

(3) -30

(4) -25

Answer 1

Correct option is (1) 35

\(\overrightarrow{\mathrm{a}}+5 \overrightarrow{\mathrm{b}}=\lambda \overrightarrow{\mathrm{c}}\)

\(\overrightarrow{\mathrm{b}}+6 \overrightarrow{\mathrm{c}}=\mu \overrightarrow{\mathrm{a}}\)

Eliminating \(\overrightarrow{a}\)

\(\lambda \overrightarrow{\mathrm{c}}-5 \overrightarrow{\mathrm{b}}=\frac{6}{\mu} \overrightarrow{\mathrm{c}}+\frac{1}{\mu} \overrightarrow{\mathrm{b}}\)

\(\therefore \mu=\frac{-1}{5}, \lambda=-30\)

\(\alpha=5, \beta=30\)