Problem 24 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(a, b, c\) are three non-parallel unit vectors such that \(a \times(b \times c)=\frac{1}{2} b\), then the angles which a makes with \(b\) and \(c\) are (A) \(90^{\circ}, 60^{\circ}\) (B) \(45^{\circ}, 60^{\circ}\) (C) \(30^{\circ}, 60^{\circ}\) (D) none of these

Step-by-Step Solution

Answer

The angles are (A) \(90^{\circ}, 60^{\circ}\).

Step 1: Understanding the Vector Triple Product

The vector triple product identity can be expressed as: \( a \times (b \times c) = (a \cdot c)b - (a \cdot b)c \). Our given condition is \( a \times(b \times c) = \frac{1}{2} b \). Substituting this into the identity, we get the equation \( (a \cdot c)b - (a \cdot b)c = \frac{1}{2} b \).

Step 2: Analyzing the Equation

Comparing the equation \( (a \cdot c)b - (a \cdot b)c = \frac{1}{2} b \) with \( yb \), it is clear that the only term containing \(b\) is \( (a \cdot c)b \). Thus, it requires that \( (a \cdot c) = \frac{1}{2} \) for the terms to match.

Step 3: Understanding the Dot Product Implication

Since \( (a \cdot c) = \frac{1}{2} \), it represents the cosine of the angle \( \theta \) between vector \(a\) and vector \(c\), i.e., \( \cos(\theta) = \frac{1}{2} \). This implies that \( \theta = 60^{\circ} \).

Step 4: Simplifying the Problem

Now consider the remaining condition fed into the equation: \(- (a \cdot b)c = 0\). As the terms are vectors, \( (a \cdot b)c \) equals the zero vector only if \( (a \cdot b) = 0 \). This implies \( cos(\phi) = 0 \), where \( \phi \) is the angle between \(a\) and \(b\).

Step 5: Solving for the Angles

From the above steps, \( a \cdot b = 0 \) indicates that the angle \(\phi = 90^{\circ}\). Meanwhile, \( a \cdot c = 0.5\) indicates \( \theta = 60^{\circ}\). Hence, the angles between \(a\) and \(b\) is \( 90^{\circ}\), and between \(a\) and \(c\) is \( 60^{\circ}\).

Key Concepts

Vector Triple Product

The vector triple product is a formula used when evaluating the cross product of one vector with the cross product of two other vectors. It involves three vectors and can be expressed with the identity:
\( a \times (b \times c) = (a \cdot c)b - (a \cdot b)c \).
This formula helps simplify complex expressions involving vector cross and dot products.

• The term \( (a \cdot c)b \) projects vector \(b\) in the direction scaled by \(a \cdot c\).
• The term \( -(a \cdot b)c \) similarly projects \(c\) in the opposite direction scaled by \(a \cdot b\).

Understanding this identity is key for tackling problems that involve multiple vector products in one expression.
In this exercise, recognizing the vector triple product allows us to set up an equation using the given vectors. This equation assists in deriving relationships and solving for angles, helping us understand the geometric orientation between vectors.

Vector Cross Product

The vector cross product is a binary operation on two vectors in three-dimensional space, resulting in a vector that is perpendicular to both of the original vectors. Given two vectors \( \mathbf{a} \) and \( \mathbf{b} \), the cross product \( \mathbf{a} \times \mathbf{b} \) is a vector.
This can be calculated using the determinant of a matrix that includes the unit vectors \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \), as well as the components of \( \mathbf{a} \) and \( \mathbf{b} \).
An important property of the cross product is its orthogonality.

• The resulting vector direction can be determined by the right-hand rule.
• The magnitude is given by \( |\mathbf{a}||\mathbf{b}| \sin(\theta) \), where \( \theta \) is the angle between \( \mathbf{a} \) and \( \mathbf{b} \).

In the context of this exercise, calculating the cross product \( a \times (b \times c) \) using the vector triple product formula simplifies to analyzing directions and projecting components, helping you derive angles effectively.

Vector Dot Product

The vector dot product is a measure of how much two vectors "point" in the same direction. It is a scalar product that results in a single number. For two vectors \( \mathbf{a} \) and \( \mathbf{b} \), the dot product is calculated as:
\( \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}| \cos(\phi) \),
where \( \phi \) is the angle between them.
This product tells you about the cosine of the angle between the two vectors and indicates whether the vectors are parallel, perpendicular, or neither.

• If \( \mathbf{a} \cdot \mathbf{b} = 0 \), the vectors are perpendicular (\( 90^{\circ} \)).
• If \( \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}| \), they are parallel and point in the same direction.

In this exercise, knowing that \( a \cdot c = \frac{1}{2} \) helped us identify an angle of \( 60^{\circ} \) between vectors \(a\) and \(c\), and \( a \cdot b = 0 \) confirmed that vectors \(a\) and \(b\) are perpendicular.