Question

If a, b and c be three non-zero vectors, no two of which are collinear. If the vector $\mathbf{a} + 2\mathbf{b}$ is collinear with c and $\mathbf{b} + 3\mathbf{c}$ is collinear with a, then ($\lambda$ being some non-zero scalar) $\mathbf{a} + 2\mathbf{b} + 6\mathbf{c}$ is equal to

• A. $\lambda\mathbf{a}$
• B. $\lambda\mathbf{b}$
• C. $\lambda\mathbf{c}$
• D. 0

Answer 1

Answer: (D)

0

Answer 2

Let $\mathbf{a} + 2\mathbf{b} = x\mathbf{c}$ and $\mathbf{b} + 3\mathbf{c} = y\mathbf{a},$ then $\mathbf{a} + 2\mathbf{b} + 6\mathbf{c} = (x + 6)\mathbf{c}$and $\mathbf{a} + 2\mathbf{b} + 6\mathbf{c} = (1 + 2y)\mathbf{a}$

So, $(x + 6)\mathbf{c} = (1 + 2y)\mathbf{a}$

Since $\mathbf{a}$ and $\mathbf{c}$ are non-zero and non-collinear, we have $x + 6 = 0$ and $1 + 2y = 0$ i.e., $x = - 6$ and $y = - \frac{1}{2}.$ In either case, we have $\mathbf{a} + 2\mathbf{b} + 6\mathbf{c} = \mathbf{0}$.