Question

Let $\vec{a} = \hat{i} + \hat{j} - \hat{k}$ and $\vec{c} = 5\hat{i} - 3\hat{j} + 2\hat{k}$ and if $\vec{b} \times \vec{c} = \vec{a}$ then $|\vec{b}|$ = ______.

• A. $\sqrt{113}$
• B. $\sqrt{114}$
• C. $\sqrt{117}$
• D. $\sqrt{119}$

Hint:

In competitive exams, if a strict interpretation of the question yields an under-defined system but the options clearly map to a simple calculation (like finding the cross product of the two given vectors), always verify if that standard calculation yields one of the exact options.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given two vectors $\vec{a}$ and $\vec{c}$. The problem text states $\vec{b} \times \vec{c} = \vec{a}$, and asks for the magnitude of $\vec{b}$.
Note: Strictly mathematically, $\vec{b} \times \vec{c} = \vec{a}$ does not uniquely define $|\vec{b}|$ (there are infinitely many such vectors). However, based on the provided options, this question commonly appears with a typo in examination papers where the intended relation was $\vec{b} = \vec{a} \times \vec{c}$. We will proceed with this highly probable intended relation to match the options.

Step 2: Key Formula or Approach:
Calculate the cross product $\vec{a} \times \vec{c}$ using determinant expansion.
Then, find the magnitude of the resulting vector using the distance formula.

Step 3: Detailed Explanation:
Assuming the intended relation is $\vec{b} = \vec{a} \times \vec{c}$:
$$\vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & -1 \\ 5 & -3 & 2 \end{vmatrix}$$
Expand the determinant along the first row:
$$\vec{b} = \hat{i} [(1)(2) - (-1)(-3)] - \hat{j} [(1)(2) - (-1)(5)] + \hat{k} [(1)(-3) - (1)(5)]$$
$$\vec{b} = \hat{i} (2 - 3) - \hat{j} (2 + 5) + \hat{k} (-3 - 5)$$
$$\vec{b} = -\hat{i} - 7\hat{j} - 8\hat{k}$$
Now, calculate the magnitude of $\vec{b}$:
$$|\vec{b}| = \sqrt{(-1)^2 + (-7)^2 + (-8)^2}$$
$$|\vec{b}| = \sqrt{1 + 49 + 64}$$
$$|\vec{b}| = \sqrt{114}$$

Step 4: Final Answer:
The magnitude is $\sqrt{114}$, which matches option (b).