Problem 84 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

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) \(\vec{c}+\left(\frac{\vec{a} \vec{c}}{\vec{a} \cdot \vec{b}}\right) \vec{b}\) (b) \(\vec{b}+\left(\frac{\vec{b} \cdot \vec{c}}{\vec{a} \vec{b}}\right) \vec{c}\) (c) \(\vec{c}-\left(\frac{\vec{a} \cdot \vec{c}}{\vec{a} \cdot \vec{b}}\right) \vec{b}\) (d) \(\vec{b}-\left(\frac{\vec{b} \cdot \vec{c}}{\vec{a} \vec{b}}\right) \vec{c}\)

Step-by-Step Solution

Answer

The vector \( \vec{d} \) is equal to option (c).

Step 1: Apply vector cross product properties

Given that \( \vec{b} \times \vec{c} = \vec{b} \times \vec{d} \), we can apply the cross product property that says if \( \vec{b} \times \vec{c} = \vec{b} \times \vec{d} \) and \( \vec{b} \) is not the zero vector, then \( \vec{c} - \vec{d} \) is parallel to \( \vec{b} \). This implies \( \vec{c} - \vec{d} = k \vec{b} \) for some scalar \( k \). Thus, \( \vec{d} = \vec{c} - k \vec{b} \).

Step 2: Use the dot product condition

Given \( \vec{a} \cdot \vec{d} = 0 \), substitute \( \vec{d} = \vec{c} - k \vec{b} \) into this equation: \( \vec{a} \cdot (\vec{c} - k \vec{b}) = 0 \). Expanding, we get \( \vec{a} \cdot \vec{c} - k(\vec{a} \cdot \vec{b}) = 0 \). Solve for \( k \): \( k = \frac{\vec{a} \cdot \vec{c}}{\vec{a} \cdot \vec{b}} \).

Step 3: Substitute the value of k

Substitute the value of \( k \) back into the expression for \( \vec{d} \): \( \vec{d} = \vec{c} - \frac{\vec{a} \cdot \vec{c}}{\vec{a} \cdot \vec{b}} \vec{b} \).

Step 4: Match with given options

Compare the derived expression for \( \vec{d} \) with the options provided. The expression \( \vec{d} = \vec{c} - \left(\frac{\vec{a} \cdot \vec{c}}{\vec{a} \cdot \vec{b}}\right) \vec{b} \) matches option (c).

Key Concepts

Cross Product

The cross product, or vector product, is a binary operation on two vectors in three-dimensional space. It results in a vector that is perpendicular to both original vectors and is often used in physics for calculations involving torques and rotations. The cross product produces a vector that aligns with the right-hand rule, meaning if you point the index finger of your right hand in the direction of the first vector (let's call it \( \vec{u} \)), and your middle finger in the direction of the second vector (\( \vec{v} \)), your thumb will point in the direction of the cross product \( \vec{u} \times \vec{v} \). Some important properties of cross products are:- If the cross product of two vectors \( \vec{b} \times \vec{c} \) equals \( \vec{b} \times \vec{d} \), and given that \( \vec{b} \) is not the zero vector, it implies that the difference \( \vec{c} - \vec{d} \) is parallel to \( \vec{b} \). This arises because the only way two non-zero vectors can have the same cross product with a common vector is if they lie along the same line when looking in the direction of that third vector.- Cross product is anti-commutative: \( \vec{u} \times \vec{v} = - (\vec{v} \times \vec{u}) \).These properties help us decipher relationships between vectors and are crucial in determining parallel vectors, as demonstrated in this exercise.

Dot Product

The dot product, also known as the scalar product, is a fundamental operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. The dot product of two vectors, \( \vec{a} \) and \( \vec{b} \), is denoted \( \vec{a} \cdot \vec{b} \) and is defined as: \[ \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 \]where \( a_1, a_2, a_3 \) are the components of \( \vec{a} \) and \( b_1, b_2, b_3 \) are the components of \( \vec{b} \).Some interesting properties include:- The dot product is distributive over addition: \( \vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} \).- The result is zero when two vectors are perpendicular (orthogonal) because their angle \( \theta = 90^\circ \), making \( \cos \theta = 0 \). - In the exercise, the condition \( \vec{a} \cdot \vec{d} = 0 \) indicates that \( \vec{d} \) is orthogonal to \( \vec{a} \).These features allow for simple calculations of vectors' angles and determining orthogonality, as used to shape our understanding of \( \vec{d} \) in the solution.

Vector Equations

Vector equations are expressions that describe geometric relationships using vectors. They are essential in fields like physics and engineering to model various phenomena. A vector equation typically includes a vector or vector operation on one side equaling another vector or operation on the other, which can often describe lines, planes, or more complex geometries.For example, one common type of vector equation represents a line in the form \( \vec{r} = \vec{a} + t \vec{b} \) where \( \vec{r} \) represents any point on the line, \( \vec{a} \) is a fixed position vector, \( \vec{b} \) is a direction vector, and \( t \) is a scalar. In the exercise, the equation \( \vec{d} = \vec{c} - k\vec{b} \) uses vector algebra to express how \( \vec{d} \) adjusts from \( \vec{c} \) due to its parallel nature to \( \vec{b} \). Combined with our solution, it helps illustrate how constraints like \( \vec{a} \cdot \vec{d} = 0 \) manipulate and define the vector \( \vec{d} \).By solving vector equations, you can determine various properties of geometrical figures, such as their intersections, angles, and dependencies among vectors.