Question

Let $\vec{a}=2\hat{i}-5\hat{j}+5\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+3\hat{k}$. If $\vec{c}$ is a vector such that $2(\vec{a}\times\vec{c})+3(\vec{b}\times\vec{c})=\vec{0}$ and $(\vec{a}-\vec{b})\cdot\vec{c}=-97$, then $|\vec{c}\times\hat{k}|^2$ is equal to

• A. 193
• B. 218
• C. 205
• D. 233

Hint:

If $(\vec{u}\times\vec{v})=\vec{0}$, then $\vec{u}$ and $\vec{v}$ are parallel — use this to reduce vector equations quickly.

Answer 1

The Correct Option is B

We need to find the value of \( |\vec{c} \times \hat{k}|^2 \) given the vectors \( \vec{a} = 2\hat{i} - 5\hat{j} + 5\hat{k} \), \( \vec{b} = \hat{i} - \hat{j} + 3\hat{k} \), and certain conditions: \( 2(\vec{a} \times \vec{c}) + 3(\vec{b} \times \vec{c}) = \vec{0} \) and \( (\vec{a} - \vec{b}) \cdot \vec{c} = -97 \).

Let's solve the problem step-by-step:

1. From the first condition, \( 2(\vec{a} \times \vec{c}) + 3(\vec{b} \times \vec{c}) = \vec{0} \), we can factor out \(\vec{c}\) and write: \(2(\vec{a} \times \vec{c}) + 3(\vec{b} \times \vec{c}) = 2 \vec{a} \times \vec{c} + 3 \vec{b} \times \vec{c} = \vec{0}\)
2. This means that \(2 \vec{a} + 3 \vec{b}\) must be proportional to \(\vec{c}\), indicating the vector \(\vec{c}\) lies in the direction of \(2 \vec{a} + 3 \vec{b}\).
3. Let's calculate the vector \(2\vec{a} + 3\vec{b}\): \(2\vec{a} = 4\hat{i} - 10\hat{j} + 10\hat{k}\) \(3\vec{b} = 3\hat{i} - 3\hat{j} + 9\hat{k}\) \[ 2\vec{a} + 3\vec{b} = (4 + 3)\hat{i} + (-10 - 3)\hat{j} + (10 + 9)\hat{k} \] \[ = 7\hat{i} - 13\hat{j} + 19\hat{k} \]
4. Using the dot product condition, \( (\vec{a} - \vec{b}) \cdot \vec{c} = -97 \): \[ \vec{a} - \vec{b} = (2 - 1)\hat{i} + (-5 + 1)\hat{j} + (5 - 3)\hat{k} \] \[ = \hat{i} - 4\hat{j} + 2\hat{k} \]
5. Assuming \(\vec{c} = \lambda(7\hat{i} - 13\hat{j} + 19\hat{k})\), calculate: \[ (\hat{i} - 4\hat{j} + 2\hat{k}) \cdot \lambda(7\hat{i} - 13\hat{j} + 19\hat{k}) = -97 \] \[ \lambda(7 \cdot 1 + (-4) \cdot (-13) + 2 \cdot 19) = -97 \] \[ \lambda(7 + 52 + 38) = -97 \] \[ \lambda \times 97 = -97 \quad \Rightarrow \quad \lambda = -1 \]
6. Thus, \(\vec{c} = - (7\hat{i} - 13\hat{j} + 19\hat{k}) = -7\hat{i} + 13\hat{j} - 19\hat{k}\).
7. We need \( |\vec{c} \times \hat{k}|^2 \): \[ \vec{c} \times \hat{k} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -7 & 13 & -19 \\ 0 & 0 & 1 \\ \end{vmatrix} = \begin{vmatrix} 13 & -19 \\ 0 & 1 \\ \end{vmatrix} \hat{i} - \begin{vmatrix} -7 & -19 \\ 0 & 1 \\ \end{vmatrix} \hat{j} + \begin{vmatrix} -7 & 13 \\ 0 & 0 \\ \end{vmatrix} \hat{k} \] \[ = (13)\hat{i} + (7)\hat{j} + 0\hat{k} = 13\hat{i} + 7\hat{j} \]
8. Finally, calculate \( |\vec{c} \times \hat{k}|^2 \): \[ |\vec{c} \times \hat{k}|^2 = 13^2 + 7^2 = 169 + 49 = 218 \]

Therefore, the value of \( |\vec{c} \times \hat{k}|^2 \) is 218.