Question

Let $\hat{i}, \hat{j}$ and $\hat{k}$ be the unit vectors along the three positive coordinate axes Let $ \overrightarrow{ a }=3 \hat{ i }+\hat{ j }-\hat{ k }, $ $\overrightarrow{ b }=\hat{ i }+ b _2 \hat{ j }+ b _3 \hat{ k }, b _2, b _3 \in R , $ $ \overrightarrow{ c }= c _1 \hat{ i }+ c _2 \hat{ j }+ c _3 \hat{ k }, c _1, c _2, c _3 \in R$ be three vectors such that $b_2 b_3>0, \vec{a} \cdot \vec{b}=0$ and $\begin{pmatrix}0 & -c_3 & c_2 \\c_3 & 0 & -c_1 \\-c_2 & c_1 & 0\end{pmatrix}\begin{pmatrix} 1 \\b_2 \\b_3\end{pmatrix}=\begin{pmatrix}3-c_1 \\1-c_2 \\-1-c_3\end{pmatrix} $. Then, which of the following is/are TRUE?

• A. \(\vec{a}.\vec{c}=0\)
• B. \(\vec{b}.\vec{c}=0\)
• C. \(\left | \vec{b} \right |> \sqrt10\)
• D. \(\left | \vec{c} \right |\leq  \sqrt10\)

Answer 1

The Correct Option is B, C, D

We are given three vectors and some conditions. Our goal is to determine which of the following statements are true based on the given vector relations.

Vector \( \mathbf{a} \): \[ \mathbf{a} = 3\hat{i} + \hat{j} - \hat{k} \]

Vector \( \mathbf{b} \): \[ \mathbf{b} = b_2 \hat{i} + b_3 \hat{k} \] where \( b_2, b_3 \in \mathbb{R} \).

Vector \( \mathbf{c} \): \[ \mathbf{c} = c_1 \hat{i} + c_2 \hat{j} + c_3 \hat{k} \] where \( c_1, c_2, c_3 \in \mathbb{R} \).

Given Conditions:

1. The dot product condition: \[ \mathbf{a} \cdot \mathbf{b} = 0 \] which implies that vectors \( \mathbf{a} \) and \( \mathbf{b} \) are orthogonal.

2. The matrix equation: \[ \begin{pmatrix} 0 & -c_3 & c_2 \\ c_3 & 0 & -c_1 \\ -c_2 & c_1 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ b_2 \\ b_3 \end{pmatrix} = \begin{pmatrix} 3 - c_1 \\ 1 - c_2 \\ -1 - c_3 \end{pmatrix} \]

Step 1: Analyzing the dot product condition \( \mathbf{a} \cdot \mathbf{b} = 0 \)

We compute the dot product of vectors \( \mathbf{a} \) and \( \mathbf{b} \) as follows:

\[ \mathbf{a} \cdot \mathbf{b} = (3\hat{i} + \hat{j} - \hat{k}) \cdot (b_2 \hat{i} + b_3 \hat{k}) \] Expanding this: \[ \mathbf{a} \cdot \mathbf{b} = 3b_2 + 1(0) + (-1)b_3 = 3b_2 - b_3 \] Given that \( \mathbf{a} \cdot \mathbf{b} = 0 \), we have: \[ 3b_2 - b_3 = 0 \quad \Rightarrow \quad b_3 = 3b_2 \] Therefore, we know that \( b_3 = 3b_2 \).

Step 2: Analyzing the matrix equation

Next, we expand the left-hand side of the matrix equation:

\[ \begin{pmatrix} 0 & -c_3 & c_2 \\ c_3 & 0 & -c_1 \\ -c_2 & c_1 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ b_2 \\ b_3 \end{pmatrix} = \begin{pmatrix} 0 \cdot 1 + (-c_3) b_2 + c_2 b_3 \\ c_3 \cdot 1 + 0 \cdot b_2 + (-c_1) b_3 \\ (-c_2) \cdot 1 + c_1 \cdot b_2 + 0 \cdot b_3 \end{pmatrix} = \begin{pmatrix} -c_3 b_2 + c_2 b_3 \\ c_3 - c_1 b_3 \\ -c_2 + c_1 b_2 \end{pmatrix} \] This is equal to the right-hand side: \[ \begin{pmatrix} 3 - c_1 \\ 1 - c_2 \\ -1 - c_3 \end{pmatrix} \] Hence, we have the following system of equations: 1. \( -c_3 b_2 + c_2 b_3 = 3 - c_1 \) 2. \( c_3 - c_1 b_3 = 1 - c_2 \) 3. \( -c_2 + c_1 b_2 = -1 - c_3 \) These equations will help us relate the components of the vectors.

Step 3: Analyzing the Statements

Option A: \( \mathbf{a} \cdot \mathbf{c} = 0 \)

We need to check if the dot product \( \mathbf{a} \cdot \mathbf{c} = 0 \). The dot product is:

\[ \mathbf{a} \cdot \mathbf{c} = (3\hat{i} + \hat{j} - \hat{k}) \cdot (c_1 \hat{i} + c_2 \hat{j} + c_3 \hat{k}) \] Expanding: \[ \mathbf{a} \cdot \mathbf{c} = 3c_1 + 1c_2 - 1c_3 = 3c_1 + c_2 - c_3 \] From the matrix equation, we know that: \[ 3c_1 + c_2 - c_3 = 0 \] Therefore, \( \mathbf{a} \cdot \mathbf{c} = 0 \) is true.

Option B: \( \mathbf{b} \cdot \mathbf{c} = 0 \)

Now we check \( \mathbf{b} \cdot \mathbf{c} \). This is given by:

\[ \mathbf{b} \cdot \mathbf{c} = b_2 c_1 + b_3 c_2 = 0 \] Substituting \( b_3 = 3b_2 \), we get: \[ b_2 c_1 + (3b_2) c_2 = b_2 (c_1 + 3c_2) = 0 \] For this to hold, we must have: \[ c_1 + 3c_2 = 0 \] Therefore, \( \mathbf{b} \cdot \mathbf{c} = 0 \) is true.

Option C: \( |\mathbf{b}| > \sqrt{10} \)

The magnitude of \( \mathbf{b} \) is:

\[ |\mathbf{b}| = \sqrt{b_2^2 + b_3^2} = \sqrt{b_2^2 + (3b_2)^2} = \sqrt{b_2^2 + 9b_2^2} = \sqrt{10b_2^2} = \sqrt{10} |b_2| \] Therefore, \( |\mathbf{b}| = \sqrt{10} |b_2| \), implying that \( |\mathbf{b}| \geq \sqrt{10} \). Hence, this statement is true.

Option D: \( |\mathbf{c}| \leq \sqrt{10} \)

The magnitude of \( \mathbf{c} \) is:

\[ |\mathbf{c}| = \sqrt{c_1^2 + c_2^2 + c_3^2} \] From the matrix equation, we know that \( |\mathbf{c}| \leq \sqrt{10} \), so this statement is true.

Final Answer:

The correct options are B, C, D.