If A ≡ 2i + 3j, B ≡ pi + 9j and C ≡ i − j Are Collinear, Then the Value of p Is

A standard vector algebra problem, not tied to a specific exam paper.

Problem

If $A \equiv 2\hat{i}+3\hat{j}$, $B \equiv p\hat{i}+9\hat{j}$, and $C \equiv \hat{i}-\hat{j}$ are collinear, then the value of $p$ is?

Key insight. Three points are collinear exactly when the vector from one of them to a second is parallel to the vector from the same point to the third. In two dimensions, “parallel” has a simple algebraic test: the determinant formed from the two vectors’ components must be zero.

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Approach

Form the vectors $\vec{AB}$ and $\vec{AC}$ by subtracting position vectors. For collinearity, these two vectors must be parallel, which in two dimensions means their components satisfy $x_1y_2-x_2y_1=0$. Solve the resulting linear equation for $p$.

Solution

Step 1 — Compute vector AB

$$\vec{AB} = B-A = (p-2)\hat{i}+(9-3)\hat{j} = (p-2)\hat{i}+6\hat{j}$$

Step 2 — Compute vector AC

$$\vec{AC} = C-A = (1-2)\hat{i}+(-1-3)\hat{j} = -\hat{i}-4\hat{j}$$

Step 3 — Apply the parallel-vectors condition

For $\vec{AB}$ and $\vec{AC}$ to be parallel (a requirement for $A$, $B$, $C$ to be collinear), their components must satisfy:

$$(p-2)(-4) – (6)(-1) = 0$$

Step 4 — Solve for p

$$-4(p-2)+6 = 0 \implies -4p+8+6 = 0 \implies -4p+14=0$$

$$p = \frac{14}{4} = \frac{7}{2}$$

Answer

Common mistakes

• Using the wrong pair of vectors for the parallel test. Any two of $\vec{AB}$, $\vec{AC}$, $\vec{BC}$ work equally well, but they must be computed consistently — mixing up which point is subtracted from which flips a sign and changes the answer.
• Setting up the determinant condition with components in the wrong order. The condition $x_1y_2-x_2y_1=0$ (not $x_1y_2+x_2y_1=0$ or with $x$ and $y$ swapped) is what correctly tests for parallel vectors — double-checking against a known collinear/non-collinear example helps catch this.

Practise next

• If $P \equiv 3\hat{i}-\hat{j}$, $Q \equiv q\hat{i}+5\hat{j}$, and $R \equiv -\hat{i}+2\hat{j}$ are collinear, find $q$, using the same parallel-vectors determinant method.