Problem 51 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A vector of magnitude \(\sqrt{51}\) which makes equal angles with the vectors \(a=\frac{1}{3}(i-2 j+2 k)\), \(b=\frac{1}{5}(-4 i-3 k)\) and \(c=j\) is given by (A) \(5 i-j-5 k\) (B) \(-5 i+j+5 k\) (C) \(5 i+j+5 k\) (D) none of these

Step-by-Step Solution

Answer

The correct answer is (D) none of these.

Step 1: Identifying the Direction Cosines

When a vector makes equal angles with three given vectors, the direction cosines of the vectors are proportional to these vectors' components. Here the vector makes equal angles with vectors \(a\), \(b\), and \(c\). That means the direction cosines \(l, m, n\) are equal, assuming the vector is \( \vec{v} = l\vec{a} + m\vec{b} + n\vec{c} \).

Step 2: Calculate the Magnitude Relation

Since the vector has a given magnitude of \( \sqrt{51} \), and the vector \( \vec{v} \) makes equal angles, it is proportional and can be expressed in terms of direction cosines as \( \vec{v} = m \cdot \vec{a}' + m \cdot \vec{b}' + m \cdot \vec{c}' \) where \(m\) is the common magnitude factor and each \(\vec{a}', \vec{b}', \vec{c}'\) are unit vectors along \(\vec{a}, \vec{b}, \vec{c}\).

Step 3: Compute Vector Components

Normalize the given vectors: \(\vec{a}, \vec{b},\) and \(\vec{c}\) become, \(\vec{a}' = \frac{1}{3}(i - 2j + 2k)\), \(\vec{b}' = \frac{1}{5}(-4i - 3k)\), and \(\vec{c}' = j\). Since \( \vec{v} = x(i - 2j + 2k) + y(-4i - 3k) + zj \) where \( x = y = z \) due to equal angles, compute further.

Step 4: Equation for Magnitude of Resultant Vector

To find the common factor \(m\), use the magnitude condition: \[ |\vec{v}| = \sqrt{51} = m \sqrt{\left( \frac{1}{3} \right)^2 (1^2 + (-2)^2 + 2^2) + \left( \frac{1}{5} \right)^2 (-4^2 + 0 + (-3)^2) + 1^2} \].

Step 5: Simplify and Solve

Simplify the combinations: substitute equality of components, and solve for \(m\) by equating the magnitude. After substituting, equating, and simplifying, we find \(m = 5\) and the resultant vector is suitably \(5i - 5j - 5k\).

Step 6: Choose the Correct Option

Check the resulting vector against the available options: From solving, \(\vec{v} = 5i - 5j - 5k\), which corresponds with the option \(B) \ -5i+j+5k\) when accounting for direction and equality by angles.

Key Concepts

Direction Cosines

When we talk about direction cosines of a vector, we are essentially referring to the angles that the vector makes with the coordinate axes. These are typically represented by the terms \( l, m, n \), which are the cosines of the angles that a vector makes with the x, y, and z axes, respectively. In the context of this exercise, we are exploring the case where a vector makes equal angles with three other vectors.

• If a vector makes equal angles with different vectors, its direction cosines are proportional to the components of those vectors.
• When given vectors \( a, b, \) and \( c \), you can assume a relationship like \( \vec{v} = l\vec{a} + m\vec{b} + n\vec{c} \).
• In problems involving equal angles, you often set \( l, m, \) and \( n \) to be equal to explore the symmetry of the problem.

Understanding direction cosines aids in visualizing how a vector is oriented in 3D space. It provides crucial insight into the geometric arrangement of vectors, making analysis and solution representation mathematically manageable.

Magnitude of a Vector

The magnitude of a vector is a measure of its length or size. In mathematics, it’s commonly derived using the Pythagorean theorem for its components. To find the magnitude of a vector \( \vec{v} = ai + bj + ck \), we use:
\[ |\vec{v}| = \sqrt{a^2 + b^2 + c^2} \]Given the problem's magnitude, \( \sqrt{51} \), we want to ensure that any combination of our vectors meets this condition:

• The relationship \( |\vec{v}| = \sqrt{51} \) gives us a benchmark to verify if a vector combination is correct.
• This is why the solution manipulates components and solves for magnitudes using proportions.
• In problems, checking magnitude guarantees the solution satisfies both directionality and size.

Remember, the magnitude imposes constraints that guide towards the correct vector solutions, ensuring the calculations remain geometrically sound.

Unit Vectors

Unit vectors are vectors with a magnitude of 1. They are primarily used to indicate direction without having any influence over the magnitude of the resultant vector. They are denoted with a "hat" notation, such as \( \hat{i}, \hat{j}, \hat{k} \), representing the x, y, and z axes directions, respectively.

• Converting any vector to a unit vector involves dividing it by its magnitude, which is crucial for proportional analysis.
• In exercises, unit vectors simplify complex expressions by focusing purely on direction.
• Given vectors can be expressed as unit vectors, we'll transform vectors \( a, b, c \) to unit forms \( \vec{a}', \vec{b}', \vec{c}' \).

In the solution, this transformation helps compare and combine direction cosines accurately. Always ensure vectors are normalized to maintain consistency in direction-based problems.