Question

If 2 vectors \( 4\hat{i} + \ell \hat{j} - 6 \hat{k} \) and \( -6 \hat{i} + 12 \hat{j} + 9 \hat{k} \) are collinear, find \(\ell\).

Hint:

For collinearity, two vectors are scalar multiples of each other, meaning their corresponding components are proportional.

Answer 1

Step 1: Use the condition for collinearity of vectors.
For two vectors to be collinear, one must be a scalar multiple of the other. That is, for vectors \(\mathbf{A}\) and \(\mathbf{B}\), we have: \[ \mathbf{A} = \lambda \mathbf{B} \]
Step 2: Set up the equation.
The given vectors are: \[ \mathbf{A} = 4\hat{i} + \ell \hat{j} - 6 \hat{k}, \quad \mathbf{B} = -6 \hat{i} + 12 \hat{j} + 9 \hat{k} \] Equating \(\mathbf{A}\) and \(\lambda \mathbf{B}\), we get the system of equations: \[ 4 = \lambda(-6), \quad \ell = \lambda(12), \quad -6 = \lambda(9) \]
Step 3: Solve for \(\ell\).
From the third equation: \[ \lambda = \frac{-6}{9} = -\frac{2}{3} \] Substitute \(\lambda = -\frac{2}{3}\) into the second equation: \[ \ell = \left(-\frac{2}{3}\right) \times 12 = -8 \] Thus, the value of \(\ell\) is: \[ \boxed{-8} \]