Vector Algebra
Coplanar Vectors
Grade 12
Question:
<p>The vectors <span class="math">x\mathbf{i} + (x+1)\mathbf{j} + (x+2)\mathbf{k}</span>, <span class="math">(x+3)\mathbf{i} + (x+4)\mathbf{j} + (x+5)\mathbf{k}</span> and <span class="math">(x+6)\mathbf{i} + (x+7)\mathbf{j} + (x+8)\mathbf{k}</span> are coplanar if <span class="math">x</span> is equal to</p>
<p>(a) 1</p>
<p>(b) -3</p>
<p>(c) 4</p>
<p>(d) 0</p>
Step-by-Step Solution
Key Concept: Three vectors are coplanar if the determinant of their direction coefficients equals zero. Column operations can reveal linearly dependent rows.
Solution: For the vectors to be coplanar, the determinant of their coefficients must be zero: \[\begin{vmatrix} x & x+1 & x+2 \\ x+3 & x+4 & x+5 \\ x+6 & x+7 & x+8 \end{vmatrix} = 0\] Applying \(C_2 \to C_2 - C_1\) and \(C_3 \to C_3 - C_1\): \[\begin{vmatrix} x & 1 & 2 \\ x+3 & 1 & 2 \\ x+6 & 1 & 2 \end{vmatrix} = 0\] Since columns 2 and 3 are identical (after the row differences), the determinant is 0 for all values of \(x\). Hence, \(x \in \mathbb{R}\)
Correct Answer: Multiple