Problem 55 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(a\) and \(b\) be two non-collinear unit vectors. If \(u=a-(a \cdot b) b\) and \(v=a \times b\), then \(|v|\) is (A) \(|u|\) (B) \(|u|+|u \cdot a|\) (C) \(|u|+|u \cdot b|\) (D) \(|u|+u \cdot(a+b)\)

Step-by-Step Solution

Answer

(A) |u|

Step 1: Use Properties of Dot and Cross Products

Given two non-collinear unit vectors \(a\) and \(b\), recall that the dot product \(a \cdot b = \cos\theta\) where \(\theta\) is the angle between \(a\) and \(b\). Since \(a\) and \(b\) are unit vectors, \(|a| = 1\) and \(|b| = 1\). The magnitude of the cross product is \(|v| = |a \times b| = \sin\theta\).

Step 2: Analyze Vector u

The vector \(u\) is given by \(u = a - (a \cdot b) b\). This operation projects \(a\) onto \(b\) and subtracts this projection from \(a\), making \(u\) perpendicular to \(b\). Since \(u\) is the component of \(a\) perpendicular to \(b\), \(|u| = \sqrt{1 - (a \cdot b)^2} = \sin\theta\).

Step 3: Compare Magnitudes of u and v

We established that \(|u| = \sin\theta\). For the cross product \(v = a \times b\), its magnitude is \(|v| = \sin\theta\). Therefore, \(|v| = |u|\).

Step 4: Determine the Correct Option

Since \(|v| = |u|\), the answer is \((A)\, |u|\). Check the options: Option \((A)\) \(|u|\) matches our finding that \(|v| = |u|\). None of the other options include \(|v| = |u|\) without additional what isn't present in the solution we've derived.

Key Concepts

Understanding Unit Vectors

Unit vectors are fundamental in vector mathematics. A unit vector is a vector that has a magnitude of 1 and is often used to indicate the direction of a vector without affecting its magnitude. This is essential as it allows the representation of direction in space irrespective of the scale. For example, if you have a vector \( vector \), the unit vector in the direction of \( vector \) is typically denoted as \( \hat{vector} \).

• Magnitudes: Every unit vector has a magnitude of 1. This makes unit vectors a standardized representation of direction.
• Examples: Common examples of unit vectors are \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) in three-dimensional coordinate systems representing directions along the x, y, and z axes, respectively.

In our exercise, vectors \( a \) and \( b \) are unit vectors with magnitudes equal to 1. This property is critical when performing dot and cross product calculations.

Exploring Vector Projection

The concept of vector projection is crucial in understanding the relationships between vectors in a given space. Projecting one vector onto another illustrates how much of one vector goes in the direction of another vector.
The projection of vector \( a \) onto vector \( b \) is calculated by the formula:\[\text{proj}_b(a) = \frac{a \cdot b}{b \cdot b} b\]In our exercise, vector \( u = a - (a \cdot b) b \), which is derived by subtracting the projection of \( a \) onto \( b \) from \( a \) itself. This subtraction leaves a component that is perpendicular to \( b \).

• Orthogonal Components: This operation allows vector \( u \) to be orthogonal to vector \( b \).
• Usage: Vector projection is often utilized in physics and engineering to resolve forces or motions along specific directions.

Calculating the Magnitude of a Vector

The magnitude of a vector is a measure of its length, regardless of its direction. When working with vectors, calculating the magnitude is a common operation used to understand the scale of vector quantities.
The magnitude of a vector \( a \), denoted as \( |a| \), is calculated using the formula:\[|a| = \sqrt{a_x^2 + a_y^2 + a_z^2}\]If the vector is expressed in different components. In unit vector terms, \( |a| = 1 \) since a unit vector's entire purpose is indicating direction.
In the provided exercise, both vectors \( a \) and \( b \) are unit vectors, meaning \( |a| = 1 \) and \( |b| = 1 \). This simplifies many calculations. For example, the magnitude of the cross product \( v = a \times b \) is \( |v| = \sin\theta \). Similarly, \( |u| = \sqrt{1 - (a \cdot b)^2} = \sin\theta \) which equals \( |v| \). Hence, understanding these magnitudes helps in determining relationships and outcomes in vector operations.