Problem 60 - Algebra for JEE ADVANCED

Question

If \(l_{1}^{2}+m_{1}^{2}+n_{1}^{2}=1\), etc. and \(l_{1} l_{2}+m_{1} m_{2}+n_{1} n_{2}=0\), etc. and \(\Delta=\left|\begin{array}{lll}l_{1} & m_{1} & n_{1} \\ l_{2} & m_{2} & n_{2} \\ l_{3} & m_{3} & n_{3}\end{array}\right|\) then a. \(|\Delta|=3\) b. \(|\Delta|=2\) c. \(|\Delta|=1\) d. \(\Delta=0\)

Step-by-Step Solution

Answer

The determinant \(|\Delta| = 1\).

Step 1: Understand the given conditions

The conditions provided are \( l_{1}^{2} + m_{1}^{2} + n_{1}^{2} = 1 \), \( l_{2}^{2} + m_{2}^{2} + n_{2}^{2} = 1 \), \( l_{3}^{2} + m_{3}^{2} + n_{3}^{2} = 1 \) which imply that each of the rows represents a unit vector. The condition \( l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2} = 0 \) implies that the first two vectors are orthogonal. Similar conditions follow for the other pairs of rows.

Step 2: Rewrite the matrix in context

Given the matrix \[\begin{bmatrix} l_{1} & m_{1} & n_{1} \ l_{2} & m_{2} & n_{2} \ l_{3} & m_{3} & n_{3} \end{bmatrix}\], it represents a matrix whose rows are mutually orthogonal unit vectors.

Step 3: Identify the nature of the matrix

Since the rows are orthogonal unit vectors in 3D space, they form a standard orthonormal basis. An orthonormal matrix has determinant \( |\Delta| = 1 \) or \(-1\).

Step 4: Calculate the determinant

The determinant of a matrix whose rows (or columns) form an orthonormal basis of the space is either \(1\) or \(-1\). Since we have three orthogonal unit vectors as rows and they span the space, the determinant \( \Delta \) is either \(+1\) or \(-1\).

Step 5: Conclude the determinant value

Since the determinant magnitude \(|\Delta|\) must satisfy a positive orientation (a typical convention unless specified is positive upon consideration of normal vectors), the possible value here considering the options given is \(|\Delta| = 1\).

Key Concepts

Unit Vectors

A unit vector is a vector that has a magnitude of 1. In three-dimensional space, a vector \( \mathbf{v} = (x, y, z) \) is a unit vector if \( \sqrt{x^2 + y^2 + z^2} = 1 \). This essentially means that each component of the vector contributes to a total length of 1.
Unit vectors are essentials in defining directions without magnitudes acting as pure directional indicators in a vector space. They are often used:

• To represent any vector as a product of a scalar and a unit vector.
• In coordinate systems to define the direction of axes, like \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) in Cartesian coordinates.

Any arbitrary vector can be expressed as a scalar multiple of a unit vector by dividing the original vector by its magnitude.

Orthogonal Vectors

Orthogonal vectors are vectors that meet at a right angle, meaning they are perpendicular to one another. The dot product of two orthogonal vectors is zero. Suppose you have vectors \( \mathbf{a} = (a_1, a_2, a_3) \) and \( \mathbf{b} = (b_1, b_2, b_3) \), they are orthogonal if \( a_1 b_1 + a_2 b_2 + a_3 b_3 = 0 \).

• Orthogonality is an essential concept in vector spaces as it simplifies the decomposition of vectors and computation of vector projections.
• Orthogonal vectors lead to orthogonal matrices where rows and columns are unit vectors and orthogonal to each other.

In practice, you can think of orthogonal vectors like two streets meeting at a right-angled corner. Such mathematical relationships are crucial for computations in physics and engineering, where they simplify the analysis of forces and dimensions.

Determinant of a Matrix

The determinant is a scalar value derived from a square matrix. It provides insights about the matrix such as whether it is invertible or not, and what volume scaling factor its transformation represents. For a 3x3 matrix, such as\[\begin{bmatrix}l_1 & m_1 & n_1 \l_2 & m_2 & n_2 \l_3 & m_3 & n_3\end{bmatrix}\]the determinant can help determine linear independence of vectors and even the orientation of a system.

• If the determinant is zero, the vectors are linearly dependent and the matrix is not invertible.
• If the determinant is either \(1\) or \(-1\), particularly in orthonormal matrices, it suggests an orthonormal basis.

When the determinant of a matrix representing an orthonormal basis is \(1\) or \(-1\), it confirms the preservation of angles and lengths in the transformation, ensuring that vectors remain orthogonally aligned.