Problem 42 - A Complete Resource Book in Mathematics for JEE Main

Question

The sides of a parallelogram are \(2 i+4 j-5 k\) and \(i+2 j\) \(+3 k\). The unit vector parallel to one of the diagonals size is (A) \(\frac{1}{7}(3 i+6 j-2 k)\) (B) \(\frac{1}{7}(3 i-6 j-2 k)\) (C) \(\frac{1}{7}(-3 i+6 j-2 k)\) (D) \(\frac{1}{7}(3 i+6 j+2 k)\)

Step-by-Step Solution

Answer

The correct option is (A) \(\frac{1}{7}(3i + 6j - 2k)\).

Step 1: Understand the Problem

We need to find a unit vector that is parallel to one of the diagonals of a parallelogram whose sides are represented by vectors \(2i + 4j - 5k\) and \(i + 2j + 3k\). A diagonal can be found by adding the vectors of the adjacent sides.

Step 2: Find the Vector of the Diagonal

For a parallelogram, one of the diagonals is the sum of the vectors of its adjacent sides. Therefore, calculate the sum of \(2i + 4j - 5k\) and \(i + 2j + 3k\): \[(2i + 4j - 5k) + (i + 2j + 3k) = (2 + 1)i + (4 + 2)j + (-5 + 3)k = 3i + 6j - 2k.\] This gives us the vector \(3i + 6j - 2k\) for the diagonal.

Step 3: Find the Magnitude of the Diagonal Vector

Calculate the magnitude of the vector \(3i + 6j - 2k\):\[\|3i + 6j - 2k\| = \sqrt{3^2 + 6^2 + (-2)^2} = \sqrt{9 + 36 + 4} = \sqrt{49} = 7.\]

Step 4: Calculate the Unit Vector

To find the unit vector in the direction of \(3i + 6j - 2k\), divide the vector by its magnitude:\[\text{Unit vector} = \frac{1}{7}(3i + 6j - 2k).\]

Step 5: Choose the Correct Option

Compare the unit vector \(\frac{1}{7}(3i + 6j - 2k)\) with the given options. The option that matches is (A) \(\frac{1}{7}(3i + 6j - 2k)\).

Key Concepts

Parallelogram Diagonals

Parallelograms are four-sided polygons with opposite sides that are equal in length and parallel. A unique feature of parallelograms is the way their diagonals intersect each other. To understand why the diagonals are relevant, remember that any parallelogram has two diagonals, each acting as a line of symmetry that divides the shape into two congruent triangles.

• The diagonal can be found by adding vectors, as each side of the parallelogram can be represented with a vector. These vectors represent both direction and magnitude, corresponding to the sides of the parallelogram.
• In the exercise, the sides of the parallelogram are given as vectors, which simplifies finding one diagonal, calculated as the sum of these vectors.

Adding these vectors together aids in determining the whole diagonal vector, giving us insight into the thorough understanding of this geometric entity within vector algebra.

Unit Vector

A unit vector is a special type of vector in mathematics and physics that has a magnitude of exactly 1 unit, making its purpose purely directional.

• To compute a unit vector, you divide any non-zero vector by its magnitude. For example, if you have a vector \( \mathbf{v} \), the unit vector \( \mathbf{u} \) in the direction of \( \mathbf{v} \) is calculated as: \( \mathbf{u} = \frac{\mathbf{v}}{\|\mathbf{v}\|} \).
• This concept is critical for simplifying vector operations and ensuring consistency across various applications since it maintains only the direction of the original vector without considering its length.
• In our exercise, after determining the diagonal vector of the parallelogram, we turned it into a unit vector, focusing purely on direction, crucial for comparing it with given options.

Understanding the unit vector ensures you're capable of expressing vectors without altering their direction, making comparisons and calculations inherently easier.

Vector Addition

Vector addition is the operation of finding a resultant vector by summing two or more vectors. The resulting vector represents the combined effect of the initial ones.

• The process begins with aligning vectors head-to-tail or tail-to-head, producing a new vector from the starting point to the endpoint.
• Mathematically, this process is facilitated by adding the corresponding components of the vectors, such as in the form \( \mathbf{a} + \mathbf{b} = (a_1 + b_1, a_2 + b_2, a_3 + b_3) \).
• In vector algebra, particularly in exercises involving geometric shapes like parallelograms, vector addition is key to establishing crucial sub-elements, such as their diagonals.

The skillful application of vector addition in the original exercise enabled the identification of a diagonal and ultimately, its unit vector, demonstrating how basic concepts intertwine to solve complex problems.