Problem 30 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let the vectors \(\vec{a}, \vec{b}, \vec{c}\) be such that \(|\vec{a}|=2,|\vec{b}|=4\) and \(|\vec{c}|=4\). If the projection of \(\vec{b}\) on \(\vec{a}\) is equal to the projection of \(\vec{c}\) on \(\vec{a}\) and \(\vec{b}\) is perpendicular to \(\vec{c}\), then the value of \(|\vec{a}+\vec{b}-\vec{c}|\) is

Step-by-Step Solution

Answer

The value of \(|\vec{a} + \vec{b} - \vec{c}|\) is 6.

Step 1: Understand Vector Projections

The projection of vector \( \vec{b} \) onto vector \( \vec{a} \) is given by \( \text{proj}_{\vec{a}} \vec{b} = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2} \vec{a} \). Similarly, the projection of \( \vec{c} \) onto \( \vec{a} \) is \( \text{proj}_{\vec{a}} \vec{c} = \frac{\vec{a} \cdot \vec{c}}{|\vec{a}|^2} \vec{a} \). We are given that these projections are equal.

Step 2: Equate the Projections

Since the projections are equal, we have \[ \frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2} \vec{a} = \frac{\vec{a} \cdot \vec{c}}{|\vec{a}|^2} \vec{a} \]. This simplifies to \( \vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} \).

Step 3: Consider Perpendicular Vectors

We know that \( \vec{b} \) is perpendicular to \( \vec{c} \), so \( \vec{b} \cdot \vec{c} = 0 \). This gives us a clue about their mutual orientation.

Step 4: Find Magnitude of the Resultant Vector

We need to find \(|\vec{a} + \vec{b} - \vec{c}|\). First calculate \((\vec{b} - \vec{c})^2\). Recalling \(\vec{b} \cdot \vec{c} = 0\), we have: \(|\vec{b} - \vec{c}|^2 = \vec{b}\cdot\vec{b} + \vec{c}\cdot\vec{c} = 4^2 + 4^2 = 32\). Since \(\vec{a}\cdot\vec{b} = \vec{a}\cdot\vec{c}\), their combined part in the resultant squared expression will be nullified. Thus, \(|\vec{a} + \vec{b} - \vec{c}| = \sqrt{|\vec{a}|^2 + 32} = \sqrt{2^2 + 32} = \sqrt{36} = 6\).

Step 5: Verify the Calculation Logging

Ensure that all calculations align with the given conditions: projections equality \( \vec{a}\cdot\vec{b} = \vec{a}\cdot\vec{c} \), and vector perpendicularity \( \vec{b}\cdot\vec{c} = 0 \). Since our calculations satisfy these, the solution is consistent with the problem statement.

Key Concepts

Dot Product

The dot product is a fundamental operation in vector algebra. It takes two vectors and returns a single scalar quantity. The mathematical expression for the dot product of two vectors \( \vec{a} \) and \( \vec{b} \) is \( \vec{a} \cdot \vec{b} = |\vec{a}| \, |\vec{b}| \, \cos \theta \), where \( \theta \) is the angle between the two vectors.

• The dot product encapsulates the idea of projecting one vector onto another.
• If the dot product is zero, the vectors are perpendicular (more on that later).
• This operation significantly aids in determining vector orientation and magnitude relations.

In the context of vector projections, the dot product plays a crucial role. To find the projection of \( \vec{b} \) onto \( \vec{a} \), we use the formula: \[\text{proj}_{\vec{a}} \vec{b} = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2} \vec{a}.\]Here, the dot product \( \vec{a} \cdot \vec{b} \) gives us a sense of how much of \( \vec{b} \) lies in the direction of \( \vec{a} \).
This concept was crucial in equating the projections of \( \vec{b} \) and \( \vec{c} \) onto \( \vec{a} \) in the exercise, emphasizing their scalar relationship.

Magnitude of Vectors

Understanding the magnitude of a vector is key to mastering vectors as it gives the 'length' of the vector. For a vector \( \vec{v} \), the magnitude (or length) is denoted by \( |\vec{v}| \). It's calculated using the square root of the sum of the squares of its components.

• For a 2D vector \( \vec{v} = \langle v_1, v_2 \rangle \): \( |\vec{v}| = \sqrt{v_1^2 + v_2^2} \).
• For a 3D vector \( \vec{v} = \langle v_1, v_2, v_3\rangle \): \( |\vec{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2} \).

In the given exercise, knowing the magnitudes of vectors \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \) helped simplify calculations. The magnitude of a resultant vector such as \( |\vec{a} + \vec{b} - \vec{c}| \) can be found through careful calculation, considering component-wise addition or subtraction and using the Pythagorean theorem where appropriate.
This concept provided crucial insight into solving for the resultant magnitude at the problem's conclusion.

Perpendicular Vectors

Perpendicular vectors have a unique relationship: they meet at a 90-degree angle. This means that their dot product is zero, as \( \cos \theta = 0 \) makes \( \vec{a} \cdot \vec{b} = 0 \) when vectors \( \vec{a} \) and \( \vec{b} \) are perpendicular.

• This property is extremely helpful in simplifying calculations.
• It confirms mutual orthogonality without needing to measure angles directly.
• Knowing vectors are perpendicular helps verify projections and orientations.

In this exercise, confirming that vectors \( \vec{b} \) and \( \vec{c} \) are perpendicular \((\vec{b} \cdot \vec{c} = 0)\) was critical. It ensured that interactions between \( \vec{b} \) and \( \vec{c} \) do not interfere with the calculations of the resultant vector \( |\vec{a} + \vec{b} - \vec{c}| \).
Perpendicularity played a pivotal role in simplifying the expression and understanding vector relationships.