Problem 15 - A Complete Resource Book in Mathematics for JEE Main

Question

If the vectors \(a\) and \(b\) are perpendicular to each other, then a vector \(v\) in terms of \(a\) and \(b\) satisfying the equations \(v \cdot a=0, v \cdot b=1\) and \([v a b]=1\) is (A) \(\frac{1}{|b|^{2}} b+\frac{1}{|a \times b|^{2}} a \times b\) (B) \(\frac{b}{|b|}+\frac{a \times b}{|a \times b|^{2}}\) (C) \(\frac{b}{|b|^{2}}+\frac{a \times b}{|a \times b|}\) (D) none of these

Step-by-Step Solution

Answer

Option (A) is the correct choice.

Step 1: Understand Given Conditions

We are given that vectors \(a\) and \(b\) are perpendicular. This implies that \(a \cdot b = 0\). We also have additional conditions for a vector \(v\):1. \(v \cdot a = 0\)2. \(v \cdot b = 1\)3. \([vab] = 1\), which refers to the scalar triple product \(v \cdot (a \times b) = 1\).

Step 2: Analyzing Condition v · a = 0

The condition \(v \cdot a = 0\) implies that vector \(v\) must be perpendicular to \(a\). Since \(a\) and \(b\) are already perpendicular, one component of \(v\) could potentially be along \(b\) because \(b\)'s dot product with \(a\) is 0 (since they are perpendicular).

Step 3: Analyzing Condition v · b = 1

This condition \(v \cdot b = 1\) directly tells us that the projection of \(v\) onto \(b\) is 1. Therefore, \(v\) must contain a component along \(b\) and its coefficient should be \(\frac{1}{|b|^2} b\) to satisfy that the dot product with \(b\) equals 1.

Step 4: Analyzing Condition [vab] = 1

The scalar triple product \([vab] = v \cdot (a \times b) = 1\) ensures that \(v\) also has a component along \(a \times b\). Since \(b\) and \(a \times b\) are perpendicular to each other, we can consider adding a vector along \(a \times b\), specifically \(\frac{a \times b}{|a \times b|^2}\) to satisfy the triple product condition.

Step 5: Construct the Vector v

By adding the components derived:- The component \(\frac{1}{|b|^2}b\) satisfies \(v \cdot b = 1\).- The component \(\frac{a \times b}{|a \times b|^2}\) ensures that \([vab] = 1\).Thus, vector \(v = \frac{b}{|b|^2} + \frac{a \times b}{|a \times b|^2}\).

Step 6: Match with Options

Compare the derived expression for \(v\) with the given options:- Option (B): \(\frac{b}{|b|} + \frac{a \times b}{|a \times b|^2}\) does not match.- Option (C): \(\frac{b}{|b|^2} + \frac{a \times b}{|a \times b|}\) is not an exact match.- Option (A): \(\frac{b}{|b|^2} + \frac{a \times b}{|a \times b|^2}\) matches.Therefore, option (A) is correct.

Key Concepts

Understanding the Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector algebra. It is an operation that takes two vectors and returns a scalar value. If you have two vectors, say \( \mathbf{a} = (a_1, a_2, a_3) \) and \( \mathbf{b} = (b_1, b_2, b_3) \), the dot product is calculated as follows:
\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \]
Here, the result is a single number, which geometrically represents the product of the magnitudes of the two vectors and the cosine of the angle between them.

• When the dot product is zero, it indicates that the vectors are perpendicular (orthogonal).
• If the dot product is positive, the angle is less than 90 degrees.
• If it's negative, the angle is greater than 90 degrees.

In the problem, the conditions \( v \cdot a = 0 \) and \( v \cdot b = 1 \) are examples of dot products that specify the relationship between the vector \( v \) and vectors \( a \) and \( b \).

Understanding the Scalar Triple Product

The scalar triple product is a way to find the volume of the parallelepiped formed by three vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \). It is denoted as \( [ \mathbf{a} \mathbf{b} \mathbf{c} ] \) or \( \mathbf{a} \cdot ( \mathbf{b} \times \mathbf{c} ) \). This operation gives a scalar value and signifies a 3D volume:
\[ \mathbf{a} \cdot ( \mathbf{b} \times \mathbf{c} ) \]
The value can be positive or negative depending on the orientation of the vectors, but the magnitude remains the same. This property is useful in determining if vectors are coplanar:

• When the scalar triple product is zero, the vectors are coplanar—lying on the same plane.
• When it's non-zero, it indicates a volume in 3D space.

In the discussed problem, the condition \( [vab] = 1 \) exemplifies the use of scalar triple product, ensuring the vector \( v \) maintains a specified angular and scalar relationship with \( a \) and \( b \).

Understanding the Cross Product

The cross product, also called the vector product, is an operation between two vectors that results in another vector. This new vector is perpendicular to the plane formed by the original vectors. For two vectors \( \mathbf{a} \) and \( \mathbf{b} \), the cross product \( \mathbf{a} \times \mathbf{b} \) is defined as:
\[ \mathbf{a} \times \mathbf{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1) \]
This vector has a direction determined by the right-hand rule, and its magnitude is equal to the area of the parallelogram formed by \( \mathbf{a} \) and \( \mathbf{b} \).

• The cross product is zero when the vectors are parallel (or one is a scalar multiple of the other).
• It has maximum magnitude when the vectors are perpendicular.

In the exercise, the term \( \frac{a \times b}{|a \times b|^2} \) highlights the cross product's role in ensuring the perpendicular nature of the resulting vector component in the calculated vector \( v \). This ensures that \( v \) aligns in 3D space according to the problem's conditions.