Problem 49 - Algebra for JEE ADVANCED

Question

Let \(\vec{a}_{r}=x \hat{i}+y, \hat{j}+z, \hat{k}, r=1,2,3\) be three mutually perpendicular unit vectors, then the value of \(\left|\begin{array}{lll}x_{1} & x_{2} & x_{3} \\ y_{1} & y_{2} & y_{3} \\ z_{1} & z_{2} & z_{3}\end{array}\right|\) is equal to a. zero b. \(\pm 1\) c. \(\pm 2\) d. none of these

Step-by-Step Solution

Answer

The value is \( \pm 1 \), option b.

Step 1: Understand the Problem

We have three mutually perpendicular unit vectors denoted as \( \vec{a}_1, \vec{a}_2, \vec{a}_3 \), and we need to find the determinant of a given 3x3 matrix formed by their components.

Step 2: Analyze Perpendicularity and Unit Property

Since the vectors are mutually perpendicular and are unit vectors, they can be expressed as \( \vec{a}_1 = (x_1, y_1, z_1) \), \( \vec{a}_2 = (x_2, y_2, z_2) \), \( \vec{a}_3 = (x_3, y_3, z_3) \). The unit vector property implies that \( \lVert \vec{a}_r \rVert = 1 \) for each \( r \). Perpendicularity implies that the dot products \( \vec{a}_1 \cdot \vec{a}_2 = \vec{a}_2 \cdot \vec{a}_3 = \vec{a}_3 \cdot \vec{a}_1 = 0 \).

Step 3: Matrix Representation

The given determinant is \( \left| \begin{array}{ccc} x_1 & x_2 & x_3 \ y_1 & y_2 & y_3 \ z_1 & z_2 & z_3 \end{array} \right| \), which represents the volume of a parallelepiped defined by the vectors \( \vec{a}_1, \vec{a}_2, \vec{a}_3 \).

Step 4: Evaluate the Determinant of Perpendicular Vectors

Since the vectors are mutually perpendicular, the volume they enclose is actually 1, because each vector corresponds to an axis in a unit cube, and the volume of the cube is 1. In mathematical terms, the determinant represents \( \pm \lVert \vec{a}_1 \rVert \lVert \vec{a}_2 \rVert \lVert \vec{a}_3 \rVert \cdot \sin(90^{\circ}) \), which simplifies to \( \pm 1 \).

Step 5: Conclusion

The determinant \( \left| \begin{array}{ccc} x_1 & x_2 & x_3 \ y_1 & y_2 & y_3 \ z_1 & z_2 & z_3 \end{array} \right| \) equals \( \pm 1 \) if the vectors are mutually perpendicular and unit vectors. Therefore, the correct answer is option b: \( \pm 1 \).

Key Concepts

Mutually Perpendicular Vectors

When we say vectors are mutually perpendicular, we mean that each vector is at a right angle (90 degrees) to the others. This geometric relationship is crucial in understanding spaces and structures, such as in physics and engineering.
In the context of vectors \( \vec{a}_1, \vec{a}_2, \vec{a}_3 \), mutually perpendicular vectors imply that:

• The dot product between any pair is zero, i.e., \( \vec{a}_1 \cdot \vec{a}_2 = 0 \), \( \vec{a}_2 \cdot \vec{a}_3 = 0 \), \( \vec{a}_3 \cdot \vec{a}_1 = 0 \).
• This is because the dot product of two vectors results in a scalar that is proportional to the cosine of the angle between them. As the angle is 90 degrees, cosine of 90 degrees is zero.

Understanding this property is key to working with vectors in spaces where right angles are significant, such as in Cartesian planes where x, y, and z axes are mutually perpendicular.

Unit Vectors

Unit vectors are vectors with a length or magnitude of exactly one unit. They are often used to indicate direction in a coordinate system without affecting magnitude measurements.
The notation \( \vec{a} = (x, y, z) \) as a unit vector implies that the length of the vector is one:

• Mathematically, this is expressed as \( \lVert \vec{a} \rVert = 1 \).
• Unit vectors in three-dimensional space commonly include \( \hat{i}, \hat{j}, \hat{k} \)—representing base vectors along x, y, and z axes, respectively.

Their role is pivotal in simplifying vector calculations, especially when establishing direction is more important than magnitude, such as in vector normalizations or in determining projections.

Vector Dot Product

The dot product, also known as the scalar product, is a way to multiply two vectors that returns a scalar. It is computed as \( \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 \) when the vectors are represented in component form.
The dot product is essential because:

• It provides a measure of how much one vector extends in the direction of another.
• It is zero for perpendicular vectors, a critical property used to check orthogonality (right-angle condition).
• It is fundamental in calculating the angle between vectors via the cosine rule, \( \vec{a} \cdot \vec{b} = \lVert \vec{a} \rVert \lVert \vec{b} \rVert \cos \theta \).

The dot product is widely used in physics and engineering, including in work calculations where force is applied in the direction of movement.

Parallelepiped Volume

A parallelepiped is a three-dimensional figure with six parallelogram faces. The volume of a parallelepiped can be calculated using the scalar triple product of three vectors forming its edges.
This is equivalent to the determinant of a matrix formed by these vectors:

• For vectors \( \vec{a}_1, \vec{a}_2, \vec{a}_3 \), the volume is \( V = |\vec{a}_1 \cdot (\vec{a}_2 \times \vec{a}_3)| \).
• In matrix form, if these vectors make up the rows of a matrix, the volume is given by the absolute value of its determinant, \( \left| \begin{array}{ccc} x_1 & x_2 & x_3 \ y_1 & y_2 & y_3 \ z_1 & z_2 & z_3 \end{array} \right| \).

For mutually perpendicular unit vectors, this determinant equals \( \pm 1 \), representing a unit cube. This underlines the significance of the determinant in understanding geometrical and spatial properties.