Problem 49 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A unit vector \(a\) makes an angle \(\frac{\pi}{4}\) with \(i\) and \(\frac{\pi}{3}\) with \(j\). If the angle between \(a\) and \(k\) is \(\theta\), where \(\theta\) is acute, then (A) \(a=\frac{1}{2} i+\frac{1}{\sqrt{2}} j+\frac{1}{2} k\) (B) \(a=\frac{1}{\sqrt{2}} i+\frac{1}{2} j+\frac{1}{2} k\) (C) \(\theta=\frac{\pi}{3}\) (D) \(\theta=\frac{\pi}{6}\)

Step-by-Step Solution

Answer

The correct choices are A and C: \(a = \frac{1}{2} i + \frac{1}{\sqrt{2}} j + \frac{1}{2} k\) and \(\theta = \frac{\pi}{3}\).

Step 1: Express the Vector in Terms of its Components

A unit vector \(a\) in 3D space is expressed as \(a = x i + y j + z k\) where \(x, y, z\) are the components along the \(x, y, z\)-axes respectively. Since \(a\) is a unit vector, we know \(\sqrt{x^2 + y^2 + z^2} = 1\), or equivalently, \(x^2 + y^2 + z^2 = 1\).

Step 2: Apply the Given Angles

The vector \(a\) makes an angle \(\frac{\pi}{4}\) with the \(i\) or \(x\)-axis. The cosine of the angle gives the projection along the axis, so \(\cos\left(\frac{\pi}{4}\right) = x = \frac{\sqrt{2}}{2}\).For the angle \(\frac{\pi}{3}\) with the \(j\) or \(y\)-axis, \(\cos\left(\frac{\pi}{3}\right) = y = \frac{1}{2}\).

Step 3: Use the Unit Vector Property

Use the equation for the unit vector \(x^2 + y^2 + z^2 = 1\). Substitute for \(x\) and \(y\): \[\left(\frac{\sqrt{2}}{2}\right)^2 + \left(\frac{1}{2}\right)^2 + z^2 = 1 \]Simplify: \[\frac{2}{4} + \frac{1}{4} + z^2 = 1 \Rightarrow \frac{3}{4} + z^2 = 1 \Rightarrow z^2 = \frac{1}{4}\]So, \(z = \frac{1}{2}\) or \(z = -\frac{1}{2}\). Since \(\theta\) is acute, \(z\) must be positive.

Step 4: Find the Correct Vector and Angle

From Step 3, since \(z = \frac{1}{2}\), the vector is \(a = \frac{\sqrt{2}}{2} i + \frac{1}{2} j + \frac{1}{2} k\). Now, find angle \(\theta\) that satisfies \(\cos(\theta) = z = \frac{1}{2}\). The angle \(\theta\) that satisfies \(\cos(\theta) = \frac{1}{2}\) is \(\frac{\pi}{3}\). Therefore, the correct options based on the calculations are A and C.

Key Concepts

Vector Components

When discussing vectors, especially in three-dimensional space, understanding vector components is crucial. A vector is an entity that has both magnitude and direction. In a 3D coordinate system, we often break down a vector into its components along the three perpendicular axes: usually labeled as the x, y, and z axes.

• The math expression of any vector can be represented as: \ a = x i + y j + z k, \ where \(i, j, k\) are the unit vectors along the x, y, z axes respectively.
• Here, \(x, y,\) and \(z\) are the vector components in their respective directions.

Understanding these components lets you analyze how much of the vector is pointed in each of the three directions. This becomes especially vital when solving problems in physics, engineering, and computer graphics.

For unit vectors, which are vectors with a magnitude of one, the equation \(x^2 + y^2 + z^2 = 1\) holds true, ensuring their unit length in any coordinate system.

Angles Between Vectors

Angles between vectors are fundamental in determining how two vectors relate to each other spatially. They allow us to understand their orientation in space and can be computed using the dot product formula.

• The dot product of two vectors \(a\) and \(b\) is given by: \ a \cdot b = |a||b|\cos(\theta), \ where \(\theta\) is the angle between the vectors.
• If \(a\) and \(b\) are unit vectors, this simplifies to \ \cos(\theta) = a \cdot b, \ since their magnitudes are both one.

This formula becomes handy, as it directly links the dot product and the angle between vectors, allowing an easy computation of one if the other is known.

For instance, if you know that a unit vector makes an angle of \(\pi/4\) with the x-axis, you can calculate its x-component using cosine, since the cosine of \(\pi/4\) is \(\sqrt{2}/2\). This methodology extends similarly to the other axes, which is particularly useful in exercises involving vectors in 3D.

3D Coordinate Geometry

3D coordinate geometry is a branch of mathematics that studies shapes, points, and vectors in three-dimensional space. This system encompasses three axes, namely x, y, and z axes, perpendicular to one another.

• In 3D geometry, points are expressed with three coordinates \((x, y, z)\).
• Vectors in this space are described using these coordinates as their components or directional indicators.

The fundamental rule in 3D vector geometry is the use of Pythagorean theorem in determining the magnitudes, where the overall length or magnitude \(|\,a\,|\) of a vector \(a\) is given by: \(|\,a\,| = \sqrt{x^2 + y^2 + z^2}\).

This branch of math is essential in fields requiring spatial understanding and design, like architecture, aerospace, and video game development, where representation and transformation of 3D objects are analyzed.

Unit Circle

The unit circle is a foundational concept in trigonometry and geometry. It's a circle with a radius of one and is centered at the origin of the coordinate system. The unit circle provides a simple way to understand trigonometric functions and their relations to angles.

• On the unit circle, any point \((x, y)\) represents \(\cos(\theta)\) and \(\sin(\theta)\) for various angles \(\theta\).
• Since the radius is one, the equations \(\cos^2(\theta) + \sin^2(\theta) = 1\) directly tie into defining the circular path.

This circle is immensely useful in solving geometrical problems and analyzing vectors. It simplifies understanding how vector angles set against coordinate axes relate to each other.

Moreover, the unit circle helps to visualize angles and trigonometric functions using radians, which is the preferred unit for many mathematical computations involving angles, especially in calculus and physics.