Problem 77 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(\overrightarrow{\left.\mathrm{c}\right|^{2}}=60\) and \(\overrightarrow{\mathrm{c}} \times(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+5 \hat{\mathrm{k}})=\overrightarrow{0}\), then a value of \(\overrightarrow{\text { c }} \cdot(-7 \hat{i}+2 \hat{j}+3 \hat{k})\) is: \(\quad\) [Online April 11, 2014] (a) \(4 \sqrt{2}\) (b) 12 (c) 24 (d) \(12 \sqrt{2}\)

Step-by-Step Solution

Answer

The value of \(\overrightarrow{c} \cdot (-7 \hat{i} + 2 \hat{j} + 3 \hat{k})\) is \(12\sqrt{2}\) (option d).

Step 1: Interpret the Given Information

We know two things: \(\overrightarrow{\mathrm{c}}\) satisfies \(\overrightarrow{\mathrm{c}} \times (\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 5\hat{\mathrm{k}}) = \overrightarrow{0}\) and \(\overrightarrow{\left|\mathrm{c}\right|^{2}}=60\). The first equation implies that \(\overrightarrow{\mathrm{c}}\) is parallel to \(\hat{\mathrm{i}} + 2 \hat{\mathrm{j}} + 5 \hat{\mathrm{k}}\).

Step 2: Write the Vector Equation

Since \(\overrightarrow{\mathrm{c}}\) is parallel to \(\hat{\mathrm{i}} + 2 \hat{\mathrm{j}} + 5 \hat{\mathrm{k}}\), it can be represented as a scalar multiple of it: \(\overrightarrow{\mathrm{c}} = k(\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 5\hat{\mathrm{k}})\).

Step 3: Use Given Magnitude

Use the modulus equation \(\left|\overrightarrow{\mathrm{c}}\right|^{2} = 60\). Plug in \(\overrightarrow{\mathrm{c}} = k(\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 5\hat{\mathrm{k}})\) to get \(k^2(1^2 + 2^2 + 5^2) = 60\).

Step 4: Solve for \(k\)

Simplify \(k^2 (1 + 4 + 25) = 60\), therefore \(k^2 \times 30 = 60\). Solve for \(k^2\) to find \(k^2 = 2\), thus \(k = \sqrt{2}\).

Step 5: Calculate the Required Dot Product

The vector \(\overrightarrow{c} = \sqrt{2}(\hat{i} + 2 \hat{j} + 5 \hat{k})\). Compute \(\overrightarrow{c} \cdot (-7 \hat{i} + 2 \hat{j} + 3 \hat{k})\) resulting in \(\sqrt{2} \cdot ((1)(-7) + (2)(2) + (5)(3))\).

Step 6: Simplify the Dot Product Expression

Calculate the expression: \((-7) + 4 + 15 = 12\). Therefore, the dot product is \(\sqrt{2} \times 12\), which simplifies to \(12\sqrt{2}\).

Key Concepts

Vector Mathematics

Vectors are essential when working with quantities that possess both magnitude and direction. Unlike scalars, which only have magnitude, vectors play a crucial role in physics and engineering to describe forces, velocities, and more.

Key characteristics of vectors include:

• Components: Vectors in a three-dimensional space can be represented as \(\overrightarrow{v} = a\hat{i} + b\hat{j} + c\hat{k}\), where \(a\), \(b\), and \(c\) are components along the \(x\), \(y\), and \(z\) axes respectively.
• Direction: The direction is usually defined by the vector's orientation in space relative to the coordinate axes.
• Addition and Subtraction: Vectors can be added or subtracted using their components. For example, adding \(\overrightarrow{v} = a\hat{i} + b\hat{j} + c\hat{k}\) and \(\overrightarrow{w} = d\hat{i} + e\hat{j} + f\hat{k}\) results in \(\overrightarrow{v} + \overrightarrow{w} = (a+d)\hat{i} + (b+e)\hat{j} + (c+f)\hat{k}\).

Understanding vectors is fundamental to solving complex problems within mathematics and the sciences, providing the language to describe physical entities accurately.

Dot Product

The dot product is an operation that takes two vectors and returns a scalar, which can be useful in determining angles between vectors or projecting one vector onto another.

The formula for the dot product is \(\overrightarrow{a} \cdot \overrightarrow{b} = a_1b_1 + a_2b_2 + a_3b_3\), where \(\overrightarrow{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}\) and \(\overrightarrow{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}\).

Main features include:

• Angle Calculation: The dot product relates to the cosine of the angle \(\theta\) between two vectors: \(\overrightarrow{a} \cdot \overrightarrow{b} = |\overrightarrow{a}|\times|\overrightarrow{b}|\times\cos\theta\).
• Orthogonality: If the dot product of two vectors is zero, it indicates that the vectors are perpendicular.

The dot product is a versatile tool in vector mathematics and is particularly helpful when determining forces and projections.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a constant, stretching or shrinking its magnitude without affecting its direction.

For a given vector \(\overrightarrow{v} = a\hat{i} + b\hat{j} + c\hat{k}\), scalar multiplication by \(k\) results in \(k\overrightarrow{v} = ka\hat{i} + kb\hat{j} + kc\hat{k}\).

Key points about scalar multiplication include:

• Effect on Magnitude: The magnitude of a vector changes by the absolute value of the scalar \(k\).
• Direction: The direction remains unchanged unless \(k\) is negative, in which case the vector is reversed.
• Transformation: Scalar multiplication is used to represent vectors as a basis or a part of linear transformations.

This process is integral to simplifying and understanding vector operations, as seen in the original exercise where the vector \(\overrightarrow{c}\) was rewritten as a scalar multiple of a known direction.

Magnitude of a Vector

The magnitude of a vector is a measure of its length. It determines how long a vector is without considering its direction.

For a vector \(\overrightarrow{v} = a\hat{i} + b\hat{j} + c\hat{k}\), the magnitude is calculated using the formula \(|\overrightarrow{v}| = \sqrt{a^2 + b^2 + c^2}\).

Important aspects of magnitude include:

• Norm: It is sometimes referred to as the norm of a vector, a measure of distance in space.
• Intuitiveness: Magnitude is analogous to length or distance in the physical world, making it a vital concept in real-world applications.
• Unit Vectors: Vectors with a magnitude of 1 are called unit vectors, commonly used to express directions.

In the exercise provided, understanding the vector's magnitude allowed the determination of the scalar \(k\), ensuring that the mathematical properties held true for the solution.