Vector Algebra
Collinearity of Points
Grade 12
Question:
<p>Let the given points be A(−1, −1, 2), B(2, m, 5) and C(3, 11, 6). Find the value of m such that A, B and C are collinear.</p>
Step-by-Step Solution
Key Concept: Three points are collinear if the vectors joining them are parallel, meaning one vector is a scalar multiple of the other.
Step 1: For collinearity, vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ must be parallel. Step 2: Calculate $\overrightarrow{AB} = (2+1)\mathbf{i} + (m+1)\mathbf{j} + (5−2)\mathbf{k} = 3\mathbf{i} + (m+1)\mathbf{j} + 3\mathbf{k}$ Step 3: Calculate $\overrightarrow{AC} = (3+1)\mathbf{i} + (11+1)\mathbf{j} + (6−2)\mathbf{k} = 4\mathbf{i} + 12\mathbf{j} + 4\mathbf{k}$ Step 4: For collinearity: $\overrightarrow{AB} = λ\overrightarrow{AC}$ From the i component: $3 = 4λ$ ⟹ $λ = 3/4$ Step 5: From the j component: $m + 1 = 12λ = 12(3/4) = 9$ ⟹ m = 8 ∴ m = 8
Correct Answer: 8