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?
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?
Updated On: Apr 23, 2024
- \(\vec{a}.\vec{c}=0\)
- \(\vec{b}.\vec{c}=0\)
- \(\left | \vec{b} \right |> \sqrt10\)
\(\left | \vec{c} \right |\leq \sqrt10\)
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The Correct Option is B, C, D
Solution and Explanation
The correct answer is option
(B) \(\vec{b}.\vec{c}=0\)
(C) \(\left | \vec{b} \right |> \sqrt10\)
(D): \(\left | \vec{c} \right |\leq \sqrt10\)
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Concepts Used:
Matrices
Matrix:
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.
The basic operations that can be performed on matrices are:
- Addition of Matrices - The addition of matrices addition can only be possible if the number of rows and columns of both the matrices are the same.
- Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
- Scalar Multiplication - The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication.
- Multiplication of Matrices - Matrices multiplication is defined only if the number of columns in the first matrix and rows in the second matrix are equal.
- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.