Question

Let \(\bar{a}=\hat{i}+2\hat{j}-3\hat{k}\) and \(\bar{a}+\bar{b}=4\hat{i}-2\hat{j}+\lambda\hat{k}\). If \(\bar{a}\cdot\bar{b}=4\), then the value of \(\lambda\) is equal to

• A. 3
• B. -3
• C. -6
• D. 6
• E. 0

Answer 1

The Correct Option is C

We are given the following vector equations: \[ \mathbf{a} = \hat{i} + 2 \hat{j} - 3 \hat{k}, \] \[ \mathbf{a} + \mathbf{b} = 4 \hat{i} - 2 \hat{j} + \lambda \hat{k}. \] First, express \( \mathbf{b} \) in terms of \( \mathbf{a} \) and the given equation: \[ \mathbf{b} = (4 \hat{i} - 2 \hat{j} + \lambda \hat{k}) - (\hat{i} + 2 \hat{j} - 3 \hat{k}). \] Simplify: \[ \mathbf{b} = (4 - 1) \hat{i} + (-2 - 2) \hat{j} + (\lambda + 3) \hat{k}, \] \[ \mathbf{b} = 3 \hat{i} - 4 \hat{j} + (\lambda + 3) \hat{k}. \] Use the dot product equation We are given \( \mathbf{a} \cdot \mathbf{b} = 4 \). Compute the dot product: \[ \mathbf{a} \cdot \mathbf{b} = (1)(3) + (2)(-4) + (-3)(\lambda + 3). \] Simplify: \[ \mathbf{a} \cdot \mathbf{b} = 3 - 8 - 3(\lambda + 3), \] \[ \mathbf{a} \cdot \mathbf{b} = -5 - 3(\lambda + 3). \] Now, substitute \( \mathbf{a} \cdot \mathbf{b} = 4 \): \[ -5 - 3(\lambda + 3) = 4. \] Solve for \( \lambda \): \[ -5 - 3\lambda - 9 = 4, \] \[ -14 - 3\lambda = 4, \] \[ -3\lambda = 18, \] \[ \lambda = -6. \]

The correct option is (C) : \(-6\)

Answer 2

We are given \(\bar{a} = \hat{i} + 2\hat{j} - 3\hat{k}\) and \(\bar{a} + \bar{b} = 4\hat{i} - 2\hat{j} + \lambda\hat{k}\). We also know that \(\bar{a} \cdot \bar{b} = 4\).

First, we can find \(\bar{b}\) by subtracting \(\bar{a}\) from \(\bar{a} + \bar{b}\):

\(\bar{b} = (\bar{a} + \bar{b}) - \bar{a} = (4\hat{i} - 2\hat{j} + \lambda\hat{k}) - (\hat{i} + 2\hat{j} - 3\hat{k}) = (4-1)\hat{i} + (-2-2)\hat{j} + (\lambda - (-3))\hat{k}\)

\(\bar{b} = 3\hat{i} - 4\hat{j} + (\lambda + 3)\hat{k}\)

Now, we can use the dot product \(\bar{a} \cdot \bar{b} = 4\):

\(\bar{a} \cdot \bar{b} = (\hat{i} + 2\hat{j} - 3\hat{k}) \cdot (3\hat{i} - 4\hat{j} + (\lambda + 3)\hat{k}) = (1)(3) + (2)(-4) + (-3)(\lambda + 3) = 4\)

\(3 - 8 - 3\lambda - 9 = 4\)

\(-5 - 3\lambda - 9 = 4\)

\(-14 - 3\lambda = 4\)

\(-3\lambda = 18\)

\(\lambda = -6\)

Therefore, the value of \(\lambda\) is -6.