Board 2026 Class 12 Maths Q14 Solution: Value of m for three collinear points

The question

The value of m for which the points with position vectors $-\hat i - \hat j + 2\hat k$ ⁠, $2\hat i + m\hat j + 5\hat k$ and $3\hat i + 11\hat j + 6\hat k$ are collinear, is

(A) $8$

(B) $-8$

(C) $2$

(D) $\frac{5}{2}$

Exam answer

Let the points be A, B and C.

\begin{aligned} \overrightarrow{AB} &= 3\hat i + (m + 1)\hat j + 3\hat k \\ \overrightarrow{AC} &= 4\hat i + 12\hat j + 4\hat k \end{aligned}

Why?

The vector from one point to another is the position vector of the end minus that of the start:

\overrightarrow{AB} = \vec b - \vec a

For $\overrightarrow{AB}$ ⁠: $2 - (-1) = 3$ ⁠, $m - (-1) = m + 1$ ⁠, $5 - 2 = 3$ ⁠.

For $\overrightarrow{AC}$ ⁠: $3 - (-1) = 4$ ⁠, $11 - (-1) = 12$ ⁠, $6 - 2 = 4$ ⁠.

For collinear points, $\overrightarrow{AB} \parallel \overrightarrow{AC}$ ⁠, so the ratios $\frac{3}{4}$ ⁠, $\frac{m + 1}{12}$ ⁠, $\frac{3}{4}$ are equal:

Why?

$\overrightarrow{AB}$ and $\overrightarrow{AC}$ both start at A. If they are parallel, A, B and C lie on one line. Parallel vectors have proportional components:

\frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3}

The first and last ratios are already $\frac{3}{4}$ ⁠, so the middle one must be $\frac{3}{4}$ too.

\frac{m + 1}{12} = \frac{3}{4} \Rightarrow m + 1 = 9

Why?

Cross-multiplying, $4(m + 1) = 3 \times 12 = 36$ ⁠, so $m + 1 = 9$ ⁠.

\begin{aligned} \frac{m + 1}{12} &= \frac{3}{4} \\ 4(m + 1) &= 3 \times 12 \\ 4(m + 1) &= 36 \\ m + 1 &= \frac{36}{4} = 9 \end{aligned}

$\therefore\ m = 8$

Why?

Take 1 from both sides: $m = 9 - 1 = 8$ ⁠.

Check: $\overrightarrow{AB} = 3\hat i + 9\hat j + 3\hat k = \frac{3}{4}\overrightarrow{AC}$ ⁠.

Why?

With $m = 8$ ⁠, the middle part of $\overrightarrow{AB}$ is $8 + 1 = 9$ ⁠. Multiply each part of $\overrightarrow{AC}$ by $\frac{3}{4}$ ⁠:

\begin{aligned} \tfrac{3}{4}(4) &= 3 \\ \tfrac{3}{4}(12) &= 9 \\ \tfrac{3}{4}(4) &= 3 \end{aligned}

Answer: (A) $8$

Where this comes from

Chapter 10, Vector Algebra: Collinear points using vectors.

Question type: Collinear points

How to spot it: click to see

Type 3. Collinear points: two pieces in the same direction

Spot it: "Show that the points are collinear", that is, they lie on one straight line.

Here: Call the points A, B, C and find $\overrightarrow{AB} = 3\hat i + (m + 1)\hat j + 3\hat k$ and $\overrightarrow{AC} = 4\hat i + 12\hat j + 4\hat k$ ⁠. Their parts must be in proportion, so $\frac{m + 1}{12} = \frac34$ ⁠.

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Q14: Value of mmm for three collinear points

Q14: Value of $m$ for three collinear points