Question

If the vectors \(2\hat{i} - 3\hat{j} + 4\hat{k}\) and \( p\hat{i} + 6\hat{j} - 8\hat{k} \) are collinear, then find the value of \( p \).

Hint:

For two vectors to be collinear, their corresponding components must be proportional by the same scalar \( k \).

Answer 1

Step 1: Condition for Vectors to be Collinear
Two vectors \( \mathbf{A} \) and \( \mathbf{B} \) are collinear if there exists a scalar \( k \) such that: \[ \mathbf{B} = k \mathbf{A} \] Step 2: Writing the Vectors in Component Form
Given the vectors: \[ (2\hat{i} - 3\hat{j} + 4\hat{k}) \quad {and} \quad (p\hat{i} + 6\hat{j} - 8\hat{k}) \] Equating the corresponding components: \[ p = 2k, \quad 6 = -3k, \quad -8 = 4k \] Step 3: Solving for the Scalar \( k \)
From the equation \( 6 = -3k \), solving for \( k \): \[ k = -2 \] Step 4: Solving for \( p \)
Substitute \( k = -2 \) into the equation \( p = 2k \): \[ p = 2(-2) = -4 \]