Question

The scalar product of the vector \( \mathbf{a} = \hat{i} - \hat{j} + 2\hat{k} \) with a unit vector along sum of vectors \( \mathbf{b} = 2\hat{i} - 4\hat{j} + 5\hat{k} \) and \( \mathbf{c} = \lambda \hat{i} - 2\hat{j} - 3\hat{k} \) is equal to 1. Find the value of \( \lambda \).

Hint:

Quick Tip: The scalar product of two vectors is computed by multiplying their corresponding components and adding the results. When the question involves a unit vector, ensure to normalize the vector by dividing by its magnitude if necessary.

Answer 1

The scalar product \( \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) \) is given as 1. First, we compute the sum of vectors \( \mathbf{b} \) and \( \mathbf{c} \): \( \mathbf{b} + \mathbf{c} = (2\hat{i} - 4\hat{j} + 5\hat{k}) + (\lambda \hat{i} - 2\hat{j} - 3\hat{k}) \). Combining like terms, we get \( \mathbf{b} + \mathbf{c} = (2 + \lambda)\hat{i} - 6\hat{j} + 2\hat{k} \). Next, we compute the scalar product of \( \mathbf{a} \) and \( (\mathbf{b} + \mathbf{c}) \): \( \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = ( \hat{i} - \hat{j} + 2\hat{k} ) \cdot \left( (2 + \lambda)\hat{i} - 6\hat{j} + 2\hat{k} \right) \). Applying the distributive property of the dot product, we have \( = 1 \cdot (2 + \lambda) + (-1) \cdot (-6) + 2 \cdot 2 \). Simplifying this expression yields \( = (2 + \lambda) + 6 + 4 \), which further simplifies to \( \lambda + 12 \). We are given that this scalar product equals 1, so \( \lambda + 12 = 1 \). Solving for \( \lambda \), we find \( \lambda = 1 - 12 = -11 \). Therefore, the value of \( \lambda \) is \( \boxed{-11} \).