Problem 72 - 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 \(a, b, c\) are three non-coplanar, non-zero vectors, then \((a \cdot a) b \times c+(a \cdot b) c \times a+(a \cdot c) a \times b=[b c a] a\) Reason: If the vectors \(a, b, c\) are non-coplanar, then so are \(b \times c, c \times a, a \times b\)

Step-by-Step Solution

Answer

B: Both are true, but the reason isn't the correct explanation.

Step 1: Understand the assertion

The assertion is: if \(a, b, c\) are three non-coplanar, non-zero vectors, then \((a \cdot a) b \times c+(a \cdot b) c \times a+(a \cdot c) a \times b=[b c a] a\). This formula suggests a property involving dot and cross products of vectors. The expression on the left seems to be a linear combination of cross products, scaled by dot products.

Step 2: Break down the assertion

We need to evaluate \((a \cdot a) b \times c + (a \cdot b) c \times a + (a \cdot c) a \times b\). If the vectors \(a, b, c\) are non-coplanar, the scalar triple product \([b c a]\) is non-zero and can be calculated using determinants. The assertion states this product yields a scaled vector \([bca]a\).

Step 3: Evaluate the reason

The reason states that if vectors \(a, b, c\) are non-coplanar, then \(b \times c, c \times a, a \times b\) are also non-coplanar. Two non-zero vectors are non-coplanar if they are not parallel. The cross products of non-coplanar vectors \(a, b, c\) are also non-zero and not parallel to each other, hence non-coplanar.

Step 4: Check validity of the reason

If vectors \(a, b, c\) are non-coplanar, their cross products do not lie in the same plane (they form a normal vector to the plane formed by any two vector among them). Hence, the reason provided is true.

Step 5: Evaluate explanation relationship

Consider the relation between the assertion and the reason. The reason being true supports the conditions under which the assertion holds (non-coplanar vectors), but it is not the causal explanation for why the assertion equals \([b c a] a\). The assertion relies on properties of vector algebra, specifically the cyclic property of the scalar triple product.

Key Concepts

Cross Product Properties

Cross product is a fundamental operation between two vectors in three-dimensional space. It results in a new vector that is perpendicular to the plane formed by the original two vectors. The cross product of two vectors \( \mathbf{u} \) and \( \mathbf{v} \) is denoted as \( \mathbf{u} \times \mathbf{v} \).

Here are some important properties of the cross product:

• Magnitude: The magnitude of \( \mathbf{u} \times \mathbf{v} \) is given by \( |\mathbf{u}| |\mathbf{v}| \sin(\theta) \), where \( \theta \) is the angle between \( \mathbf{u} \) and \( \mathbf{v} \).
• Direction: The direction of the cross product follows the right-hand rule — if the fingers of your right hand curve from \( \mathbf{u} \) to \( \mathbf{v} \), your thumb points in the direction of \( \mathbf{u} \times \mathbf{v} \).
• Non-commutativity: The cross product is anti-commutative, meaning \( \mathbf{u} \times \mathbf{v} = - (\mathbf{v} \times \mathbf{u}) \).
• Zero Vector: The cross product of two parallel vectors is a zero vector (\( \mathbf{u} \times \mathbf{u} = \mathbf{0} \)).
• Distributive over Addition: \( \mathbf{u} \times (\mathbf{v} + \mathbf{w}) = \mathbf{u} \times \mathbf{v} + \mathbf{u} \times \mathbf{w} \).

Understanding these properties is crucial because they are used to determine the relationship between vectors in space, particularly when assessing vector alignment, plane perpendicularity, and in calculations that involve area of parallelograms formed by vectors.

Scalar Triple Product

The scalar triple product is an expression involving three vectors and is represented by \([\mathbf{a} \mathbf{b} \mathbf{c}]\). It equates to \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \) and results in a scalar value.

Some key aspects of the scalar triple product are:

• Geometric Interpretation: The absolute value \(|\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|\) represents the volume of the parallelepiped formed by the vectors \( \mathbf{a}, \mathbf{b}, \) and \( \mathbf{c} \).
• Cyclic Property: The scalar triple product is invariant under cyclic permutations of its vectors: \([\mathbf{a} \mathbf{b} \mathbf{c}] = [\mathbf{b} \mathbf{c} \mathbf{a}] = [\mathbf{c} \mathbf{a} \mathbf{b}]\).
• Non-zero Condition: If \([\mathbf{a} \mathbf{b} \mathbf{c}] eq 0\), then vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are linearly independent and hence non-coplanar.

In the given assertion, the scalar triple product helps establish whether vectors are non-coplanar and is critical when proving identities involving vector magnitudes and directions. Thus, it serves as a tool for determining the spatial configuration of vectors.

Non-Coplanar Vectors

Vectors are non-coplanar if they do not lie within the same plane. This is an important concept when dealing with spatial configurations.

Key factors to determine non-coplanarity include:

• Scalar Triple Product: If \([\mathbf{a} \mathbf{b} \mathbf{c}] eq 0\), then the vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are non-coplanar. This condition implies that no linear combination of the vectors can produce a zero vector unless the coefficients are all zero.
• Geometric Interpretation: Non-coplanar vectors can span a three-dimensional space. They form a parallelepiped when connected in sequence.
• Cross Products: The assertion that if vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are non-coplanar, so are \( \mathbf{b} \times \mathbf{c}, \mathbf{c} \times \mathbf{a}, \mathbf{a} \times \mathbf{b} \). This holds because the cross products result in vectors that are orthogonal to the plane defined by the respective pair of vectors.

Recognizing non-coplanar vectors is essential in vector algebra as it dictates the possibility of forming three-dimensional structures and resolving problems involving spatial dynamics.