Problem 103 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The vectors \(\vec{a}\) and \(\vec{b}\) are not perpendicular and \(\vec{c}\) and \(\vec{d}\) are two vectors satisfying: \(\vec{b} \times \vec{c}=\vec{b} \times \vec{d}\) and \(\vec{a} \cdot \vec{d}=0 .\) Then, the vector \(\vec{d}\) is equal to (A) \(c+\left(\frac{a . c}{a \cdot b}\right) b\) (B) \(b+\left(\frac{b . c}{a \cdot b}\right) c\) (C) \(c-\left(\frac{a . c}{a \cdot b}\right) b\) (D) \(b-\left(\frac{b \cdot c}{a \cdot b}\right) c\)

Step-by-Step Solution

Answer

The vector \( \vec{d} \) is equal to \( \vec{c} - \left( \frac{\vec{a} \cdot \vec{c}}{\vec{a} \cdot \vec{b}} \right) \vec{b} \).

Step 1: Analyze the Cross Product Condition

We are given that \( \vec{b} \times \vec{c} = \vec{b} \times \vec{d} \). This implies that \( \vec{b} \times ( \vec{c} - \vec{d} ) = \vec{0} \). Since the cross product is zero, \( \vec{c} - \vec{d} \) must be parallel to \( \vec{b} \). Therefore, \( \vec{d} = \vec{c} + k\vec{b} \) for some scalar \( k \).

Step 2: Apply the Dot Product Condition

We are also given \( \vec{a} \cdot \vec{d} = 0 \). Substituting \( \vec{d} = \vec{c} + k\vec{b} \) into the dot product equation gives us \( \vec{a} \cdot ( \vec{c} + k\vec{b} ) = 0 \). Expanding this results in \( \vec{a} \cdot \vec{c} + k( \vec{a} \cdot \vec{b} ) = 0 \).

Step 3: Solve for the Scalar k

From \( \vec{a} \cdot \vec{c} + k( \vec{a} \cdot \vec{b} ) = 0 \), we can solve for \( k \): \( k = -\frac{\vec{a} \cdot \vec{c}}{\vec{a} \cdot \vec{b}} \).

Step 4: Substitute k Back into the Expression for d

Substitute \( k = -\frac{\vec{a} \cdot \vec{c}}{\vec{a} \cdot \vec{b}} \) back into \( \vec{d} = \vec{c} + k\vec{b} \): \[ \vec{d} = \vec{c} - \left( \frac{\vec{a} \cdot \vec{c}}{\vec{a} \cdot \vec{b}} \right) \vec{b} \]. This expression matches option (C).

Key Concepts

Vectors

Vectors are mathematical entities that have both magnitude and direction. They are commonly used in physics and engineering to represent quantities such as force, velocity, and acceleration.

• A vector is typically represented by an arrow in space where its length denotes the magnitude and the arrow indicates the direction.
• Algebraically, vectors are represented as ordered pairs or triplets, such as \((x, y\)) in 2D space or \((x, y, z\)) in 3D space.

When working with vectors, you'll often perform operations like addition, subtraction, and multiplication.
These operations help you decompose, project, and manipulate vectors for various applications in geometry and physics.
Understanding vectors is fundamental to mastering more complex topics like dot and cross products, which involve calculating the angle between vectors and determining perpendicular components.

Dot Product

The dot product, or scalar product, is an operation that takes two vectors and returns a scalar value. This operation is useful for finding the angle between two vectors, as well as for determining if they are perpendicular.

• The formula for the dot product of two vectors \(\vec{a} \) and \(\vec{b} \) is given as \(\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta\), where \(|\vec{a}|\) and \(|\vec{b}|\) are the magnitudes of the vectors and \(\theta\) is the angle between them.
• The dot product is zero when two vectors are perpendicular, indicating that they do not share any direction.

It is essential in projecting one vector onto another, calculating work done by a force, and more applications in physics and engineering.
Understanding this concept allows you to solve problems that involve orthogonal projections and angles between vectors.

Cross Product

The cross product, also known as the vector product, is a binary operation on two vectors in three-dimensional space, resulting in a third vector that is orthogonal to the plane containing the first two.
The magnitude of the cross product is represented by the formula \( |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta \), where \(\theta \) is the angle between the two vectors.

• The resulting vector’s direction is determined by the right-hand rule, where the thumb points in the direction of the vector given that the fingers align from \(\vec{a} \) to \(\vec{b} \).
• If the cross product is zero, it implies that the vectors are parallel since their sine angle is zero, meaning the angle between them is zero or 180 degrees.

Utilizing the cross product, you can acquire insight into how vectors interact in 3D space, such as calculating the torque exerted by a force or determining the area of a parallelogram formed by two vectors.

Scalar Multiplication

Scalar multiplication is the process of multiplying a vector by a scalar (a single number).
This operation alters the magnitude of the vector but not its direction, unless the scalar is negative, which then reverses the direction.

• Mathematically, a vector \(\vec{v} = (x, y, z)\) scaled by a factor \(k\) becomes \(k\vec{v} = (kx, ky, kz)\).
• Scalar multiplication is primarily used for scaling vectors, reflecting them across axes, or changing their units.

In vector problems, scalar multiplication can simplify components and aid in solving linear equations incorporating vector quantities.
It's crucial in fields like linear algebra and physics, especially when dealing with transformations, projections, and other vector operations.