Problem 41 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(\hat{a}, \hat{b}\) and \(\hat{c}\) are unit vectors satisfying \(\hat{a}-\sqrt{3} \hat{b}+\hat{c}=\overrightarrow{0}\), then the angle between the vectors \(\hat{a}\) and \(\hat{c}\) is: |Online April 22, 2013] (a) \(\frac{\pi}{4}\) (b) \(\frac{\pi}{3}\) (c) \(\frac{\pi}{6}\) (d) \(\frac{\pi}{2}\)

Step-by-Step Solution

Answer

The angle between \( \hat{a} \) and \( \hat{c} \) is \( \pi \).

Step 1: Recognize the Vector Equation

To solve the problem, we start with the vector equation given: \( \hat{a} - \sqrt{3} \hat{b} + \hat{c} = \overrightarrow{0} \). Since this equation must hold true, it implies that \( \hat{a} + \hat{c} = \sqrt{3} \hat{b} \). This is our rearranged equation.

Step 2: Dot Product with Vector \( \hat{a} \)

Next, we take the dot product of the entire equation with vector \( \hat{a} \):\( (\hat{a} + \hat{c}) \cdot \hat{a} = (\sqrt{3} \hat{b}) \cdot \hat{a} \).The left-hand side simplifies to \( \hat{a} \cdot \hat{a} + \hat{c} \cdot \hat{a} = 1 + \hat{c} \cdot \hat{a} \).The right-hand side simplifies to \( \sqrt{3} (\hat{b} \cdot \hat{a}) \).

Step 3: Use Unit Vector Property

Since \( \hat{b} \) is a unit vector and perpendicular to \( \hat{a} + \hat{c} \) (because its coefficients must add up to the vector's result as zero), we assume \( \hat{b} \cdot \hat{a} = 0 \) because \( \hat{b} \) and \( \hat{a} + \hat{c} \) are not aligned. This assumption implies:\( 1 + \hat{c} \cdot \hat{a} = 0 \) as per derived simplifications from the perpendicular property.

Step 4: Solve for Dot Product \( \hat{c} \cdot \hat{a} \)

Given our simplified equation from Step 3 is \( 1 + \hat{c} \cdot \hat{a} = 0 \), solving gives \( \hat{c} \cdot \hat{a} = -1 \).

Step 5: Determine the Angle Between \( \hat{a} \) and \( \hat{c} \)

The dot product formula \( \hat{c} \cdot \hat{a} = \cos(\theta) \) with \( \hat{c} \cdot \hat{a} = -1 \) leads us to conclude that \( \cos(\theta) = -1 \), implying that the angle \( \theta \) is \( \pi \) radians.

Key Concepts

Vectors

Vectors are mathematical objects that have both magnitude and direction. They are widely used in physics and mathematics to represent quantities such as force, velocity, or displacement. Vectors can be visually represented as arrows, where the length of the arrow represents the magnitude and the direction of the arrow specifies the direction of the vector.
To mathematically express a vector, we often use components along the coordinate axes, like \(\vec{v} = x \hat{i} + y \hat{j} + z \hat{k}\), where \(x, y,\) and \(z\) are the vector's components along the x, y, and z axes respectively.
Vectors are essential in calculations like finding direction, projecting onto other vectors, and more. Operations involving vectors include addition, subtraction, and various products, which are foundational in understanding vector spaces and their applications.

Unit Vectors

A unit vector is a vector that has a magnitude of 1. It is often used to signify direction without affecting magnitude. Unit vectors are generally used to express other vectors in terms of their direction.
In three-dimensional space, the standard unit vectors are denoted by \( \hat{i}, \hat{j}, \hat{k} \) along the x, y, and z axes, respectively. These vectors form the basis for defining any vector in space as a linear combination of these unit vectors.

• Any vector can be converted into a unit vector by dividing it by its magnitude.
• Unit vectors are often used to simplify the calculation of dot products and projections.

Unit vectors are vital in physics and engineering because they allow precise directionality to be assigned to scalar quantities.

Angle Between Vectors

The angle between two vectors is a measure of how "far apart" the vectors are in terms of direction. To find the angle, we often use the dot product formula, which relates the angle to the cosine function.
The mathematical representation is given by the equation: \[\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta)\]Where \(\theta\) is the angle between the vectors \(\vec{a}\) and \(\vec{b}\). Solving for \(\theta\) gives:\[\theta = \cos^{-1}\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}\right)\]
Using this equation, you can find the angle if you know the dot product and magnitudes of the vectors involved.

• When vectors are perpendicular, the angle is \(\frac{\pi}{2}\) radians.
• If they are aligned in the same direction, the angle is 0, and opposite directions correspond to \(\pi\) radians.

Dot Product

The dot product, also known as the scalar product, is a mathematical operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation is denoted by a dot between two vectors.The formula for the dot product in terms of vector components is:\[\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 + \ldots\]It can also be represented using the magnitude and angle between the vectors as:\[\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta)\]Where \(\theta\) is the angle between the vectors. The dot product has several properties:

• It is commutative: \(\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}\).
• The result is a scalar quantity — not a vector.
• If the dot product of two vectors is zero, they are perpendicular.

The dot product is key to finding angles between vectors and is widely used in physics to calculate work done and projections.