Problem 90 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \((\bar{a} \times \bar{b}) \times \bar{c}=\bar{a} \times \bar{b} \times \bar{c})\) where \(\bar{a}, \bar{b}\) and \(\bar{c}\) are any three vectors such that \(\bar{a} \bar{b} \neq 0, \bar{b} \cdot \bar{c} \neq 0\) then \(\bar{a}\) and \(\bar{c}\) are [2006] (a) inclined at an angle of \(\frac{\pi}{3}\) between them (b) inclined at an angle of \(\frac{\pi}{6}\) between them (c) perpendicular (d) parallel

Step-by-Step Solution

Answer

\(\bar{a}\) and \(\bar{c}\) are perpendicular.

Step 1: Analyze Equation

Given the equation \((\bar{a} \times \bar{b}) \times \bar{c} = \bar{a} \times \bar{b} \times \bar{c}\). This equation involves a vector triple product on the left and a scalar triple product on the right.

Step 2: Vector Triple Product Identity

Recall that the vector triple product identity is \((\bar{p} \times \bar{q}) \times \bar{r} = (\bar{r} \cdot \bar{p})\bar{q} - (\bar{r} \cdot \bar{q})\bar{p}\). Apply this identity to \((\bar{a} \times \bar{b}) \times \bar{c}\).

Step 3: Apply the Identity

By applying the vector triple product identity to \((\bar{a} \times \bar{b}) \times \bar{c}\), we get \(((\bar{a} \cdot \bar{c}) \bar{b} - (\bar{b} \cdot \bar{c}) \bar{a})\).

Step 4: Compare with Right Side

The scalar triple product \(\bar{a} \times \bar{b} \times \bar{c}\) simplifies to a scalar \((\bar{a} \cdot (\bar{b} \times \bar{c})) = (\bar{b} \cdot (\bar{a} \times \bar{c}))\), which is equal to zero since it is a scalar.

Step 5: Equate Both Sides

Since both sides of the original equation should be equal, \((\bar{a} \cdot \bar{c}) \bar{b} - (\bar{b} \cdot \bar{c}) \bar{a} = 0\). This implies \((\bar{a} \cdot \bar{c}) \bar{b} = (\bar{b} \cdot \bar{c}) \bar{a}\).

Step 6: Solve the Equation

This implies \(\bar{a}(\bar{b} \cdot \bar{c}) = \bar{b}(\bar{a} \cdot \bar{c})\). Dividing both sides by \((\bar{b} \cdot \bar{c})\), \(\bar{a} = \bar{b}\), since \(\bar{b} \cdot \bar{c} eq 0\), \(\bar{a} \cdot \bar{c} = 0\), indicating \(\bar{a}\) and \(\bar{c}\) are perpendicular.

Key Concepts

Vector Triple Product

In vector algebra, the vector triple product is an operation involving the cross product of three vectors. It is denoted as \( (\bar{a} \times \bar{b}) \times \bar{c} \), which simplifies using the identity:

• \( (\bar{p} \times \bar{q}) \times \bar{r} = (\bar{r} \cdot \bar{p})\bar{q} - (\bar{r} \cdot \bar{q})\bar{p} \)

This identity allows us to rewrite the cross product of a vector's cross product in terms of dot products. It's a useful tool for simplifying calculations involving multiple vectors. The result is still a vector, and this technique can help immensely when solving complex vector equations.
Applying this identity helps reduce the complexity of the problem by expressing it in terms of simpler operations like dot products and scalar multiples.

Scala Triple Product

The scalar triple product of three vectors is a combination of the dot and cross product, typically expressed as \( \bar{a} \cdot (\bar{b} \times \bar{c}) \).
This expression calculates a scalar and is useful in determining the volume of the parallelepiped formed by the three vectors.

• If the value is zero, it indicates that the vectors are coplanar.
• The expression itself evaluates the "volume" aspect, providing critical information about vector orientations in space.

Furthermore, this can be interchangeably expressed using cyclic permutations of the dot product and cross product, owing to its properties: \( \bar{a} \cdot (\bar{b} \times \bar{c}) = \bar{b} \cdot (\bar{c} \times \bar{a}) = \bar{c} \cdot (\bar{a} \times \bar{b}) \).
This flexibility helps simplify the manipulation and comparison of the triple products within a mathematical problem.

Perpendicular Vectors

Vectors are said to be perpendicular if their dot product is zero. This means that the angle between them is 90 degrees, and they have a relationship where one is orthogonal to the other.
In the equation derivation process, showing that \( \bar{a} \cdot \bar{c} = 0 \) was crucial. It suggests that \( \bar{a} \) and \( \bar{c} \) do not influence each other's direction.

• Perpendicular vectors are often crucial in physics and engineering to resolve forces and analyze independent components within systems.
• Understanding perpendicularity aids in breaking down complex vector systems into simpler parts, where each vector acts independently of the other.

This property was key to concluding that \( \bar{a} \) and \( \bar{c} \) in the original problem statement are orthogonal, leading to the answer that the two vectors are perpendicular, which is vital for solving and understanding vector equations.