Problem 37 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(a, b\) and \(c\) are three unit vectors such that \(a+b+c\) is also a unit vector and \(\theta_{1}, \theta_{2}\) and \(\theta_{3}\) are angles between the vectors \(a, b ; b, c\) and \(c, a\), respectively, then among \(\theta_{1}, \theta_{2}\) and \(\theta_{3^{*}}\) (A) all are acute angles (B) all are right angles (C) at least one is obtuse angle (D) none of these

Step-by-Step Solution

Answer

C: At least one angle is obtuse.

Step 1: Understand Given Conditions

Given vectors \(a\), \(b\), and \(c\) are unit vectors, meaning their magnitudes are 1, i.e., \(|a| = |b| = |c| = 1\). The vector \(a + b + c\) is also a unit vector.

Step 2: Use Vector Property

The square of the magnitude of a vector can be determined using its components. Since \(a + b + c\) is a unit vector, we have \(|a + b + c|^2 = 1\).

Step 3: Expand the Vector Equation

Expand \(|a + b + c|^2 = 1\) using vector properties: \[|(a + b + c)|^2 = |a|^2 + |b|^2 + |c|^2 + 2(a \cdot b) + 2(b \cdot c) + 2(c \cdot a)\]Since \(|a| = |b| = |c| = 1\), this simplifies to:\[1 + 1 + 1 + 2(a \cdot b) + 2(b \cdot c) + 2(c \cdot a) = 1\]Which reduces to:\[3 + 2(a \cdot b + b \cdot c + c \cdot a) = 1\]

Step 4: Solve For Dot Products

The equation from the previous step becomes:\[2(a \cdot b + b \cdot c + c \cdot a) = -2\]Divide both sides by 2, we get:\[a \cdot b + b \cdot c + c \cdot a = -1\]

Step 5: Interpret Dot Product Results

The dot product \(a \cdot b = |a||b|\cos\theta_1 = \cos\theta_1\) (since \(|a| = |b| = 1\)) and similarly for the other dot products. Therefore,\[\cos\theta_1 + \cos\theta_2 + \cos\theta_3 = -1\]For this sum to be negative, at least one \(\cos\theta_i\) must be negative indicating that the corresponding angle \(\theta_i\) is obtuse.

Key Concepts

Unit Vectors

In vector algebra, unit vectors play a fundamental role. A unit vector is simply a vector with a magnitude of 1. It points in a specific direction, but has no other size but 1. This makes unit vectors useful as a basis for direction without altering the magnitudes involved.
For any vector \( \mathbf{v} = (v_1, v_2, v_3) \), its unit vector is given by:

• \( \mathbf{u} = \frac{\mathbf{v}}{|\mathbf{v}|} = \left( \frac{v_1}{|\mathbf{v}|}, \frac{v_2}{|\mathbf{v}|}, \frac{v_3}{|\mathbf{v}|} \right) \)

Since unit vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) as in our problem each have a magnitude of 1. Therefore, their direction is preserved while simplifying calculations for better understanding of vector operations.

Dot Product

The dot product, also known as the scalar product, is a method to multiply two vectors. It provides information about the angle between vectors and their directional alignment. The dot product of two vectors \( \mathbf{a} = (a_1, a_2, a_3) \) and \( \mathbf{b} = (b_1, b_2, b_3) \) is calculated as:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \)

This resulting value gives insight beyond simple multiplication. Let's note that it can be expressed in terms of magnitude and angle:

• \( \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos(\theta) \)

Where \( \theta \) is the angle between the two vectors. Importantly, if \( \mathbf{a} \cdot \mathbf{b} > 0 \), the vectors are oriented in an acute angle and if \( \mathbf{a} \cdot \mathbf{b} < 0 \), the angle is obtuse.

Magnitude of Vectors

The magnitude of a vector, often called its length or norm, is a measure of how long the vector actually is. For any vector \( \mathbf{v} = (v_1, v_2, v_3) \), its magnitude is calculated using the following equation:

• \( |\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2} \)

This formula, derived from the Pythagorean theorem in three dimensions, provides the vector's actual length regardless of its direction.
In our problem, the unit vectors \( \mathbf{a}, \mathbf{b} \), and \( \mathbf{c} \) each have magnitudes of 1, serving as a building block to simplify understanding of more complex vector operations. Knowing that \( |\mathbf{a} + \mathbf{b} + \mathbf{c}| = 1 \) guides us to explore the relationship of these vectors and the angles formed among them.

Obtuse Angles

An obtuse angle is one that lies between greater than 90 degrees and less than 180 degrees. Within the framework of vector algebra and dot products, an obtuse angle is significant as it directly influences the sign of the dot product between two vectors.
Recall that the dot product relationship is \( \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos(\theta) \). When \( \theta \) is obtuse (greater than 90 degrees), the cosine of \( \theta \) is negative, resulting in a negative dot product.

• \( \cos \theta < 0 \Rightarrow \theta \text{ is obtuse} \)

In the context of our problem, the sum \( \cos\theta_1 + \cos\theta_2 + \cos\theta_3 = -1 \) points to at least one obtuse angle, indicating a negative cosine value, thereby determining the orientation and nature of vector relations in space.