Problem 109 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three unit vectors such that \(\vec{a} \times(\vec{b} \times \vec{c})=\frac{\sqrt{3}}{2}(\vec{b}+\vec{c})\). if \(\vec{b}\) is not parallel to \(\vec{c}\), then the angle between \(\vec{a}\) and \(\vec{b}\) is (A) \(\frac{5 \pi}{6}\) (B) \(\frac{3 \pi}{4}\) (C) \(\frac{\pi}{2}\) (D) \(\frac{2 \pi}{3}\)

Step-by-Step Solution

Answer

The angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{5\pi}{6}\) (Option A).

Step 1: Vector Identity Usage

To simplify the term \(\vec{a}\times(\vec{b} \times \vec{c})\), we use the vector triple product identity, which states that for any vectors \(\vec{u}, \vec{v}, \vec{w}\), \(\vec{u} \times (\vec{v} \times \vec{w}) = (\vec{u} \cdot \vec{w})\vec{v} - (\vec{u} \cdot \vec{v})\vec{w}\). Applying this identity to our problem, we obtain: \(\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}\).

Step 2: Equate to Given Expression

According to the problem, \(\vec{a} \times (\vec{b} \times \vec{c}) = \frac{\sqrt{3}}{2}(\vec{b} + \vec{c})\). Equate the expression from the vector identity to this given expression: \((\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c} = \frac{\sqrt{3}}{2}(\vec{b} + \vec{c})\).

Step 3: Compare Coefficients

Match the coefficients of the vectors \(\vec{b}\) and \(\vec{c}\) on both sides of the equation: 1. Coefficient of \(\vec{b}\) gives: \(\vec{a} \cdot \vec{c} = \frac{\sqrt{3}}{2}\).2. Coefficient of \(\vec{c}\) gives: \(- (\vec{a} \cdot \vec{b}) = \frac{\sqrt{3}}{2}\), therefore, \(\vec{a} \cdot \vec{b} = - \frac{\sqrt{3}}{2}\).

Step 4: Find the Angle Between \(\vec{a}\) and \(\vec{b}\)

Since \(\vec{a}\) and \(\vec{b}\) are unit vectors, the dot product formula is given by: \(\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta = \cos \theta\). From the previous step, \(\cos \theta = -\frac{\sqrt{3}}{2}\). The angle \(\theta\) whose cosine is \(-\frac{\sqrt{3}}{2}\) is \(\frac{5\pi}{6}\).

Key Concepts

Vector Identity

Vectors often interact in complex ways, and understanding these interactions is crucial for solving many physics and mathematics problems. One such interaction is the **vector triple product**, which provides a formula for simplifying the expression \( \vec{a} \times (\vec{b} \times \vec{c}) \). This can be a bit tricky since it involves cross products, which themselves result in vectors perpendicular to a plane determined by the two vectors involved. The **vector triple product identity** helps to simplify these terms, making work with them more manageable. It is given by the equation: \[\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}.\]Key points to remember for using the vector identity:- The result is a vector that is a combination of the original two vectors \( \vec{b} \) and \( \vec{c} \).- The coefficients for these vectors are dot products, which represent projections or how much one vector points in the direction of another.- This identity is particularly useful because it turns a complicated triple cross product into a sum of two simpler terms.- Understanding and using this identity efficiently can be an extremely helpful skill in vector calculus, physics equations, and engineering problems.

Angle Between Vectors

Determining the angle between two vectors is a fundamental concept in vector mathematics. This angle is indicative of how the two vectors are oriented with respect to one another. To find this angle, we can use a vector operation called the dot product.The dot product of two vectors \( \vec{u} \) and \( \vec{v} \) is calculated as:\[\vec{u} \cdot \vec{v} = |\vec{u}| |\vec{v}| \cos \theta,\]where \( \theta \) is the angle between the two vectors, and \( |\vec{u}| \) and \( |\vec{v}| \) are their magnitudes, or lengths. However, when dealing with **unit vectors**, which are vectors of length 1, this simplifies to:\[\vec{u} \cdot \vec{v} = \cos \theta.\]This relationship means that the cosine of the angle can easily be found directly from the dot product if you remember that unit vectors always have a magnitude of 1. Therefore, once you know the dot product of two unit vectors, you can readily solve for the angle.

• For example, if \( \vec{a} \cdot \vec{b} = -\frac{\sqrt{3}}{2} \), then \( \cos \theta = -\frac{\sqrt{3}}{2} \).
• This corresponds to an angle of \( \theta = \frac{5\pi}{6} \).

This allows for a quick and intuitive grasp of how vectors are positioned relative to each other. Remember, the cosine function is negative in the second quadrant of the unit circle, which helps when identifying the angle direction.

Dot Product of Unit Vectors

The dot product is an essential operation in vector mathematics, especially when dealing with unit vectors. The beauty of using unit vectors is their constraint in magnitude — they always have a length of 1. This simplification makes calculations straightforward and intuitive. Here, let’s dive into how the dot product works specifically with these vectors.When computing the **dot product of unit vectors** \( \vec{u} \) and \( \vec{v} \), the formula reduces to exactly the cosine of the angle between them, since their magnitudes are 1:\[\vec{u} \cdot \vec{v} = \cos \theta,\]Source: This shows that the dot product essentially measures the extent to which one vector points in the direction of another, scaled by their lengths. For unit vectors, this measurement becomes even more pure, directly reflecting the angle factor.Some key insights include:- When \( \vec{u} \) and \( \vec{v} \) are parallel, their dot product reaches its maximum, reflecting a value of 1.- A dot product of zero indicates orthogonality, meaning the vectors are perpendicular.- If the dot product is negative, as seen in our exercise where \( \vec{a} \cdot \vec{b} = -\frac{\sqrt{3}}{2} \), it suggests that the angle is obtuse.With these properties, the dot product becomes a powerful tool to not only determine magnitudes and angles, but also to explore geometric relationships and interpretations among vectors.