Problem 73 - A Complete Resource Book in Mathematics for JEE Main

Question

In the following questions an Assertion (A) is given followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: If $$ \begin{aligned} &i \times[(a-j) \times i]+j \times[(a-k) \times j]+k \\ &\times[(a-i) \times k]=0, \text { then } \\ &a=\frac{1}{2}(i+j+k) \end{aligned} $$ Reason: \((a \cdot i) i+(a \cdot j) j+(a \cdot k) k=a\) for any vector \(a\).

Step-by-Step Solution

Answer

(A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A).

Step 1: Analyze the Assertion

The assertion states a vector identity involving the cross product and the condition when the result equals zero. Specifically, it involves the terms: \(i \times[(a-j) \times i]+j \times[(a-k) \times j]+k \times[(a-i) \times k]=0\). To solve this, expand each term using the vector triple product identity \( \mathbf{a} \times ( \mathbf{b} \times \mathbf{c} ) = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}\).

Step 2: Simplify Each Term

Simplify each cross product term using the vector triple product identity. For example, \(i \times [(a-j) \times i]\) becomes \((i \cdot i)(a-j) - (i \cdot (a-j))i = (a - j) - a_i i\) because \(i \cdot i = 1\). Similarly, simplify the other terms: \(j \times [(a-k) \times j]\) and \(k \times [(a-i) \times k]\).

Step 3: Set Up the Equation

Combine the simplified forms: \[(a - ia_i - j) + (b - ib_j - k) + (c - ic_k - i) = 0\]. This implies that each component along \(i, j, k\) axes should independently equal zero, leading to three equations.

Step 4: Solve for vector Components

From the equation \((a - ia_i), (b - jb_j), (c - kc_k)\) equals zero, find \(a = \frac{1}{2}(i + j + k)\) which fits all conditions of vector equality along the axes from this step.

Step 5: Verify the Reason

The reason claims \( (a \cdot i) i + (a \cdot j) j + (a \cdot k) k = a \). This is indeed true by projecting onto the component axes and reconstituting \(a\) using its dot products with unit vectors \(i, j, k\).

Step 6: Determine the Relationship Between Assertion and Reason

Since both the assertion and the reason hold true and the projection identity helps reaffirm the solution derived, the reason is a correct explanation of the assertion.

Key Concepts

Cross Product

The cross product is a fundamental operation in vector mathematics. It is used to find a vector that is perpendicular to two given vectors. This operation is only defined in three-dimensional space, making it a unique tool for such calculations. The cross product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \), denoted as \( \mathbf{a} \times \mathbf{b} \), is calculated using the formula:\[\mathbf{a} \times \mathbf{b} = (a_2b_3 - a_3b_2)\mathbf{i} - (a_1b_3 - a_3b_1)\mathbf{j} + (a_1b_2 - a_2b_1)\mathbf{k}\]This result is a vector that is orthogonal, meaning perpendicular, to both \( \mathbf{a} \) and \( \mathbf{b} \). Some important properties of the cross product include:

• The cross product of two parallel vectors is zero.
• The magnitude of \( \mathbf{a} \times \mathbf{b} \) is \( |\mathbf{a}| |\mathbf{b}| \sin \theta \), where \( \theta \) is the angle between the vectors.
• The direction of the result follows the right-hand rule.

Using these properties, we can interpret vector relationships and solve problems involving vector equations, such as the one in the original exercise.

Vector Identities

Vector identities are equations that hold true for any set of vectors and are used to simplify vector calculations. One key identity is the vector triple product identity, which is particularly useful for solving complex problems:\[\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}\]This identity is essential in simplifying cross product expressions like the one in the exercise. It allows us to transform a complex nested cross product into a more manageable form using dot products. For example, suppose \( \mathbf{a}, \mathbf{b}, \text{ and } \mathbf{c} \) are vectors. Applying the triple product identity will yield two simpler terms, involving separate dot products and scalar multiplications:

• The term \((\mathbf{a} \cdot \mathbf{c})\mathbf{b}\) scales vector \(\mathbf{b}\) by how much \(\mathbf{a}\) overlaps \(\mathbf{c}\).
• The term \((\mathbf{a} \cdot \mathbf{b})\mathbf{c}\) scales vector \(\mathbf{c}\) based on its overlap with \(\mathbf{a}\).

Using vector identities helps solve complex vector problems more swiftly and accurately by reducing them to simpler, familiar forms.

Dot Product

The dot product is another primary operation in vector algebra, used to find the product of two vectors along the same direction. For vectors \( \mathbf{a} \) and \( \mathbf{b} \), the dot product, denoted as \( \mathbf{a} \cdot \mathbf{b} \), is calculated as:\[\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\]This result is a scalar value, not a vector, and it's a measure of the vectors' alignment.Key characteristics of the dot product include:

• If two vectors are perpendicular, their dot product is zero.
• The dot product can be used to calculate the angle between vectors: \( \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|} \).
• The projection of one vector onto another is given by the vector \( \mathbf{p} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|^2} \mathbf{b} \).

In the context of the exercise, the dot product is used to confirm vector components and simplify solutions. By breaking down vectors into their components based on orthogonal projections and sums, results are validated efficiently.