Problem 11 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(r=\lambda(a \times b)+\mu(b \times c)+v(c \times a)\) and \([a b c]=\frac{1}{8}\), then \(\lambda+\mu+v\) is equal to (A) \(r \cdot(a+b+c)\) (B) \(8 r-(a+b+c)\) (C) \(4 r \cdot(a+b+c)\) (D) none of these

Step-by-Step Solution

Answer

\(\lambda + \mu + v = 8r\). The answer is (D) none of these.

Step 1: Understand the Expression for r

First, let's interpret the expression provided for \( r \): \[ r = \lambda(a \times b) + \mu(b \times c) + v(c \times a). \] This expression involves cross products of the vectors \( a, b, \) and \( c \) and scalar multipliers \( \lambda, \mu, \) and \( v \).

Step 2: Utilize the Given Triple Product

We are also given the scalar triple product \([a\ b\ c] = \frac{1}{8}\). Recall that the triple product is related to the volume of the parallelepiped formed by \( a, b, \) and \( c \), and can be expressed in terms of dot and cross products. It can be utilized in vector algebra to solve the problem at hand.

Step 3: Apply Vector Identities

Recall the vector identity that relates to permutations of the cross product: \( a \cdot (b \times c) + b \cdot (c \times a) + c \cdot (a \times b) = [a\ b\ c]. \) This identity suggests that the sum \( \lambda + \mu + v \) could potentially be equal to a multiple of this triple product.

Step 4: Solve for \( \lambda + \mu + v \)

Given \( r = \lambda (a \times b) + \mu (b \times c) + v (c \times a) \) and recognizing the form \( \lambda a \cdot (b \times c) + \mu b \cdot (c \times a) + v c \cdot (a \times b) = [a\ b\ c] = \frac{1}{8} \), this implies that: \[ \lambda a \cdot (b \times c) = \mu b \cdot (c \times a) = v c \cdot (a \times b). \]This means \( r \cdot (a+b+c) = \frac{1}{8}. \) Therefore, we find \( \lambda+\mu + v = 8r. \)

Step 5: Compare to Provided Options

The expression we obtain, \( \lambda + \mu + v = 8r \), should be compared with the given options to see if any match our result. It aligns directly with the option (D) which states “none of these” since no option exactly matches \( 8r \).

Key Concepts

Cross Product

The cross product is a fundamental operation in vector algebra. It involves two vectors and results in a third vector that is perpendicular to the plane containing the original vectors. This resultant vector has a magnitude equal to the area of the parallelogram that the vectors span and a direction determined by the right-hand rule.

For two vectors \( \mathbf{a} \) and \( \mathbf{b} \), the cross product is denoted as \( \mathbf{a} \times \mathbf{b} \). The formula for the cross product in a three-dimensional Cartesian coordinate system is:\[\mathbf{a} \times \mathbf{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1). \]
Some key points about the cross product include:

• It is not commutative: \( \mathbf{a} \times \mathbf{b} eq \mathbf{b} \times \mathbf{a} \). In fact, \( \mathbf{b} \times \mathbf{a} = - (\mathbf{a} \times \mathbf{b}) \).
• The cross product of parallel vectors is zero.
• The cross product can be used to determine the perpendicular direction of two vectors in 3D space.

Triple Scalar Product

The triple scalar product is an operation involving three vectors that results in a scalar. It is heavily used in vector algebra to compute volumes and is synonymous with the mixed product of vectors. The triple scalar product of vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) is defined as:\[[\mathbf{a} \ \mathbf{b} \ \mathbf{c}] = \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}).\]
This scalar value can be interpreted as the volume of the parallelepiped formed by the three vectors. If \([\mathbf{a} \ \mathbf{b} \ \mathbf{c}] = 0\), it indicates that the vectors are coplanar.
Key characteristics of the triple scalar product:

• It is invariant under cyclic permutations of vectors: \([\mathbf{a} \ \mathbf{b} \ \mathbf{c}] = [\mathbf{b} \ \mathbf{c} \ \mathbf{a}] = [\mathbf{c} \ \mathbf{a} \ \mathbf{b}].\)
• If the product is positive, it means the vector triplet follows the right-handed rule; if negative, the opposite.
• It helps in determining the directed volume of a parallelepiped.

Understanding these properties aids in problem-solving, like evaluating the equation given in the exercise.

Vector Identity

In vector algebra, identities simplify complex vector expressions and calculations. One essential identity related to the exercise is that of the vector triple product or Lagrange's identity. This can be seen in the expression:\[\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) + \mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) + \mathbf{c} \cdot (\mathbf{a} \times \mathbf{b}) = [\mathbf{a} \ \mathbf{b} \ \mathbf{c}].\]
This identity is valuable because it ties together the dot product and cross product, illustrating relationships in a more generalized form.
Using vector identities effectively aids in simplifying and solving equations more straightforwardly. For instance, in our exercise, the identity supports solving the expression relating to the sum \( \lambda + \mu + v \), determining the relationships between the terms.

• These identities reduce computational complexity.
• They allow for elegant solutions to vector equations.
• Applying identities helps in abstract reasoning across mathematics and physics.