Problem 95 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(\vec{u}, \vec{v}\) and \(\vec{w}\) are three non- coplanar vectors, then \((\vec{u}+\vec{v}-\vec{w}) \cdot(\vec{u}-\vec{v}) \times(\vec{v}-\vec{w})\) equals (a) \(3 \vec{u} \cdot \vec{v} \times \vec{w}\) (b) 0 (c) \(\vec{u} \cdot(\vec{v} \times \vec{w})\) (d) \(\vec{u} \cdot \vec{w} \times \vec{v}\).

Step-by-Step Solution

Answer

The expression simplifies to \( \vec{u} \cdot (\vec{v} \times \vec{w}) \), which is option (c).

Step 1: Identify the Expression

We start by analyzing the expression \( (\vec{u}+\vec{v}-\vec{w}) \cdot (\vec{u}-\vec{v}) \times (\vec{v}-\vec{w}) \)

Step 2: Use the Vector Triple Product Identity

The expression involves a dot product of a vector with a cross product, which suggests using the identity \( \vec{a} \cdot (\vec{b} \times \vec{c}) = \vec{b} \cdot (\vec{c} \times \vec{a}) = \vec{c} \cdot (\vec{a} \times \vec{b}) \). Set \( \vec{a} = \vec{u} + \vec{v} - \vec{w}, \vec{b} = \vec{u} - \vec{v}, \vec{c} = \vec{v} - \vec{w} \).

Step 3: Expand Using Properties of Cross Products

First expand the cross product \( (\vec{u}-\vec{v}) \times (\vec{v}-\vec{w}) \). This gives: \[\begin{align*}(\vec{u}-\vec{v}) \times (\vec{v}-\vec{w}) &= \vec{u} \times \vec{v} - \vec{u} \times \vec{w} - \vec{v} \times \vec{v} + \vec{v} \times \vec{w} \ &= \vec{u} \times \vec{v} - \vec{u} \times \vec{w} + \vec{v} \times \vec{w} \end{align*}\]

Step 4: Substitute Back into Dot Product

Now substitute this back into the original dot product term:\[(\vec{u} + \vec{v} - \vec{w}) \cdot (\vec{u} \times \vec{v} - \vec{u} \times \vec{w} + \vec{v} \times \vec{w})\]

Step 5: Distribute the Dot Product

Expand the dot product by distributing: \[\begin{align*}&= (\vec{u} + \vec{v} - \vec{w}) \cdot (\vec{u} \times \vec{v}) - (\vec{u} + \vec{v} - \vec{w}) \cdot (\vec{u} \times \vec{w}) + (\vec{u} + \vec{v} - \vec{w}) \cdot (\vec{v} \times \vec{w})\end{align*}\]Each component simplifies separately because of orthogonality and distributive properties.

Step 6: Apply Orthogonality and Non-Coplanarity

Due to non-coplanarity and orthogonality: - Dot products of the form \((\vec{x} \pm \vec{y}) \cdot (\vec{x} \times \vec{z})\) and combinations therein generally result in zero due to linearly independence. - focus particularly on terms like \(\vec{u} \cdot (\vec{v} \times \vec{w})\) which do not cancel.

Step 7: Final Simplification

Simplifying with non-zero contributions leads us to realize that the intact viable candidates from the equation include terms such as \(\vec{u} \cdot (\vec{v} \times \vec{w})\). Simplifying to such significant contributions, namely, leaves us with:\[\vec{u} \cdot (\vec{v} \times \vec{w})\]

Step 8: Compare with Given Options

Compare the result \( \vec{u} \cdot (\vec{v} \times \vec{w}) \) with the given options. The correct answer matches option (c).

Key Concepts

Cross Product

The cross product, also known as the vector product, is a binary operation on two vectors in three-dimensional space. It results in a third vector that is perpendicular to both of the original vectors. Imagine two vectors \( \vec{a} \) and \( \vec{b} \); their cross product is denoted as \( \vec{a} \times \vec{b} \). This product will be perpendicular to the plane containing \( \vec{a} \) and \( \vec{b} \).

A few key properties of the cross product include:

• The magnitude of the cross product \( |\vec{a} \times \vec{b}| \) is equal to the area of the parallelogram that \( \vec{a} \) and \( \vec{b} \) span.
• It is not commutative, which means \( \vec{a} \times \vec{b} eq \vec{b} \times \vec{a} \), in fact, \( \vec{a} \times \vec{b} = - (\vec{b} \times \vec{a}) \).
• If \( \vec{a} \) and \( \vec{b} \) are parallel, which means they point in the same or opposite directions, their cross product will be the zero vector.

The concept of cross product is instrumental in determining orthogonal vectors, which plays a critical role in the vector triple product seen in mathematical expressions like this exercise.

Dot Product

The dot product, also referred to as the scalar product, is another fundamental operation for vectors. It is used to find the product of two vectors and results in a scalar rather than a vector. Given vectors \( \vec{a} \) and \( \vec{b} \), their dot product is denoted as \( \vec{a} \cdot \vec{b} \), and calculated as \( |\vec{a}| |\vec{b}| \cos(\theta) \), where \( \theta \) is the angle between the two vectors.

Some useful properties of the dot product include:

• It is commutative, meaning \( \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} \).
• The dot product of orthogonal (perpendicular) vectors is zero because the cosine of 90 degrees is zero.
• The dot product helps to determine projections and angles between vectors, as it essentially measures how much one vector extends in the direction of another.

In the context of the exercise problem, the dot product is a crucial component, helping to translate the intersection and alignment of vectors in space, critical when combined with the cross product for non-coplanar vectors.

Non-Coplanar Vectors

Vectors are considered non-coplanar if they do not all lie in the same plane. In a three-dimensional space, this essentially means that the vectors cannot all be confined to a single surface, much like three non-parallel sticks meeting at one point.

The importance of non-coplanar vectors in vector mathematics lies in their intrinsic independence.

• Non-coplanar vectors ensure a three-dimensional span, allowing them to form a basis for 3D space.
• A set of three non-coplanar vectors can be used to describe any point within a three-dimensional space using linear combinations.
• In terms of vector operations like the vector triple product, non-coplanar vectors ensure certain results are non-zero, making them significant in multiple mathematical and physics applications.

Understanding non-coplanarity is key in problems like the one in the exercise, where the goal is to assess the interaction of vectors through operations such as dot and cross products. It establishes the spatial orientation necessary for determining outcomes related to volume and three-dimensional arrangements.