Problem 29 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The vectors \(a, b\) and \(c\) are equal in length and taken pairwise, they make equal angles. If \(a=i+j, b=j+\) \(k\), and \(c\) makes an obtuse angle with the base vector \(i\), then \(c\) is equal to (A) \(i+k\) (B) \(-i+4 j-k\) (C) \(\frac{-1}{3} i+\frac{4}{3} j-\frac{1}{3} k\) (D) \(\frac{1}{3} i+\frac{-4}{3} j+\frac{1}{3} k\).

Step-by-Step Solution

Answer

The correct vector is (C) \(\frac{-1}{3}i+\frac{4}{3}j-\frac{1}{3}k\).

Step 1: Calculate length of vectors a and b

First, calculate the magnitude of vectors \(a\) and \(b\). The vector \(a = i+j\), so its magnitude is \( \|a\| = \sqrt{1^2 + 1^2} = \sqrt{2} \). The vector \(b = j+k\), so its magnitude is \( \|b\| = \sqrt{1^2 + 1^2} = \sqrt{2} \). Both vectors have the same length, which is \(\sqrt{2}\).

Step 2: Verify angle condition between vectors a and b

Given that the vectors taken pairwise make equal angles, check the angle between vectors \(a\) and \(b\) using their dot product: \( a \cdot b = (i + j) \cdot (j + k) = 1 \times 1 + 1 \times 0 = 1 \). The magnitude of each is \(\sqrt{2}\) so the cosine of the angle between them is \( \cos \theta = \frac{a \cdot b}{\|a\|\|b\|} = \frac{1}{2} \). The angle is acute.

Step 3: Determine possible c vectors with equal length

From Step 1, we know \(\|c\| = \sqrt{2}\). List vector options: (A) \(i+k\), (B) \(-i+4j-k\), (C) \(\frac{-1}{3}i+\frac{4}{3}j-\frac{1}{3}k\), and (D) \(\frac{1}{3}i-\frac{4}{3}j+\frac{1}{3}k\). Compute their magnitudes and check for \(\sqrt{2}\).

Step 4: Calculate magnitude for each option

Calculate magnitude for each option: (A) \(\|i+k\| = \sqrt{2}\).(B) \(\|-i+4j-k\|=\sqrt{1^2+4^2+1^2} = \sqrt{18}\).(C) \(\|\frac{-1}{3}i+\frac{4}{3}j-\frac{1}{3}k\| = \sqrt{\left(\frac{-1}{3}\right)^2+\left(\frac{4}{3}\right)^2+\left(\frac{-1}{3}\right)^2} = \sqrt{\frac{18}{9}} = \sqrt{2}\).(D) \(\|\frac{1}{3}i-\frac{4}{3}j+\frac{1}{3}k\| = \sqrt{\frac{18}{9}} = \sqrt{2}\).

Step 5: Confirm the obtuse angle criterion with i

Since vector \(c\) should form an obtuse angle with \(i\), calculate the dot product: - For (A) \(i+k\), \((i+k) \cdot i = 1\) (not obtuse).- For (C) \(\frac{-1}{3}i+\frac{4}{3}j-\frac{1}{3}k\), \(\left(\frac{-1}{3}i+\frac{4}{3}j-\frac{1}{3}k\right) \cdot i = \frac{-1}{3}\) (obtuse).- For (D) \(\frac{1}{3}i+\frac{-4}{3}j+\frac{1}{3}k\), \(\left(\frac{1}{3}i+\frac{-4}{3}j+\frac{1}{3}k\right) \cdot i = \frac{1}{3}\) (not obtuse).

Step 6: Choose the correct vector c

Based on the calculations, (C) \(\frac{-1}{3}i+\frac{4}{3}j-\frac{1}{3}k\) satisfies both the equal magnitude and the obtuse angle condition.

Key Concepts

Understanding Vector Magnitude

The concept of vector magnitude is fundamental in vector mathematics. Imagine a vector as a directed line segment in space with both magnitude, or length, and direction. To calculate the magnitude of a vector, which can be thought of as its "size," we utilize the formula for the Euclidean norm. For a vector represented as \( \mathbf{v} = (v_1, v_2, v_3) \), its magnitude is computed as \( \| \mathbf{v} \| = \sqrt{v_1^2 + v_2^2 + v_3^2} \). This formula is akin to finding the hypotenuse of a right triangle using the Pythagorean theorem.
For example, in the provided exercise, vector \(a = i + j\) means \(i\) and \(j\) both have component values of 1 along the x and y axes, respectively. Calculating its magnitude involves finding \( \|a\| = \sqrt{1^2 + 1^2} = \sqrt{2} \). Calculating vector magnitude helps identify whether vectors have the same length, which is crucial for determining geometric relationships like parallelism or similarity among vectors.
Whenever working with vectors, always start with understanding their magnitude. This foundational step aids further in applications such as comparing vectors or resolving more complex vector operations, such as the dot product.

Exploring the Dot Product

The dot product, also known as the scalar product, is a key operation between two vectors that yields a single number, or scalar. This product gives insight into the geometric relationship between the vectors, particularly their alignment or angle relative to each other. To compute the dot product of two vectors \( \mathbf{u} = (u_1, u_2, u_3) \) and \( \mathbf{v} = (v_1, v_2, v_3) \), use the formula: \( \mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3 \).
A valuable property of the dot product is its relation to the angle \( \theta \) between two vectors: \( \mathbf{u} \cdot \mathbf{v} = \| \mathbf{u} \| \| \mathbf{v} \| \cos \theta \). If the dot product is positive, the angle is acute; if negative, it is obtuse, and if zero, the vectors are perpendicular.
In the exercise, to verify the angle condition between vectors \(a\) and \(b\), we calculated: \((i + j) \cdot (j + k) = 1\). Here, since \( \cos \theta = \frac{1}{2} \), it indicates an acute angle. Understanding the dot product is integral for determining orthogonality, perpendicularity, and angles between vectors, making it an essential tool in vector mathematics.

Angles Between Vectors

Angles between vectors provide valuable geometric information. Knowing whether an angle is acute, right, or obtuse reveals the directional relationship between vectors, a crucial aspect in physics and engineering applications. The cosine of the angle \( \theta \) between two vectors can be calculated using the dot product formula: \( \cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|} \). An understanding of these angles allows for insights into vector interactions and the potential formation of vector bases.
In the given problem, it's necessary to determine that vector \(c\) makes an obtuse angle with the base vector \(i\). This is achieved by ensuring that the dot product of \(c\) with \(i\) is negative, indicating that these vectors point in opposite directions relative to each other. The vector \(c = \frac{-1}{3}i+\frac{4}{3}j-\frac{1}{3}k\) satisfies this condition since \(c \cdot i = \frac{-1}{3}\).
Mastering the angles between vectors is crucial for fields requiring optimization of paths, forces, and systems. It helps in various scenarios, including maximizing efficiency in transportation, aerodynamic modeling, and optimizing structural designs.