Problem 54 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If sdaa \(\vec{a}, \vec{b}, \vec{c}\) are vectors such that \(\vec{a}+\vec{b}+\vec{c}=0\) and \(|\vec{a}|=7,|\vec{b}|=5,|\vec{c}|=3\) then angle between vector \(\vec{b}\) and \(\rightarrow\) \(c\) is (a) \(60^{\circ}\) (b) \(30^{\circ}\) (c) \(45^{\circ}\) (d) \(90^{\circ}\)

Step-by-Step Solution

Answer

The angle between vectors \(\vec{b}\) and \(\vec{c}\) is \(60^{\circ}\) (Option a).

Step 1: Understand the given condition

We are given that \(\vec{a} + \vec{b} + \vec{c} = 0\). This implies \(\vec{a} = - (\vec{b}+\vec{c})\). This helps us express one vector in terms of the others.

Step 2: Apply the condition to the magnitudes

We also have the magnitudes: \(|\vec{a}| = 7\), \(|\vec{b}| = 5\), and \(|\vec{c}| = 3\). With \(\vec{a} = - (\vec{b}+\vec{c})\), we can write \(|\vec{a}| = |-(\vec{b} + \vec{c})| = |\vec{b} + \vec{c}|\).

Step 3: Use magnitude relationship

By substituting the magnitudes, we have \(|\vec{b} + \vec{c}| = 7\). Use the identity \(|\vec{b} + \vec{c}|^2 = |\vec{b}|^2 + |\vec{c}|^2 + 2 \vec{b} \cdot \vec{c}\) and substitute the known values to find the dot product \(\vec{b} \cdot \vec{c}\).

Step 4: Solve for the dot product

Insert the known magnitudes into the equation: \(7^2 = 5^2 + 3^2 + 2 \vec{b} \cdot \vec{c}\). Simplifying, we have: \(49 = 25 + 9 + 2 \vec{b} \cdot \vec{c}\). This leads to \(49 = 34 + 2 \vec{b} \cdot \vec{c}\).

Step 5: Simplify to find the dot product

From \(49 = 34 + 2 \vec{b} \cdot \vec{c}\), we find that \(2 \vec{b} \cdot \vec{c} = 15\), so \(\vec{b} \cdot \vec{c} = \frac{15}{2}\).

Step 6: Determine the cosine of the angle

Using the equation \(\vec{b} \cdot \vec{c} = |\vec{b}| |\vec{c}| \cos \theta\), we plug in the values: \(\frac{15}{2} = 5 \times 3 \times \cos \theta\).

Step 7: Solve for \(\cos \theta\)

From \(\frac{15}{2} = 15 \cos \theta\), it follows that \(\cos \theta = \frac{1}{2}\).

Step 8: Identify the angle

The angle \(\theta\) for which \(\cos \theta = \frac{1}{2}\) is \(60^{\circ}\).

Key Concepts

Vector Algebra

Vector algebra is a key concept in mathematics and physics that involves the study and manipulation of vectors. Vectors are entities that have both magnitude and direction, and they can represent various quantities, such as force or velocity. Understanding vector algebra is essential for solving problems involving multiple dimensions, where direction and magnitude play a crucial role.

Vectors can often be added or subtracted to generate new vectors. For instance, if \(\vec{a}, \vec{b},\) and \(\vec{c}\) are vectors such that their sum is zero: \(\vec{a} + \vec{b} + \vec{c} = 0\), it implies \(\vec{a}\) is equivalent to the negative sum of the other two vectors: \(\vec{a} = - (\vec{b} + \vec{c})\). By rearranging vectors and using algebraic identities, you can fully understand the relationship between different vectors and solve complex problems.

• Basic operations include addition, subtraction, and scalar multiplication of vectors.
• The expression \(\vec{a} = - (\vec{b} + \vec{c})\) shows how vectors can be rearranged.
• Vector algebra helps in determining unknown vectors using known conditions.

Angle Between Vectors

The angle between two vectors is an important concept that gives insight into the orientation of one vector with respect to another. It is determined using the cosine of the angle formed by the vectors. The formula used to find the cosine of the angle \(\theta\) between two vectors \(\vec{b}\) and \(\vec{c}\) is derived from the dot product formula:\[\vec{b} \cdot \vec{c} = |\vec{b}| |\vec{c}| \cos \theta\]To calculate the angle accurately, follow these steps:

• First, determine the dot product of the vectors.
• Then, find the magnitudes of each vector.
• Use the dot product and magnitudes to solve for the cosine of the angle.

The value of \(\cos \theta\) helps in deducing the angle. For example, if \(\cos \theta = \frac{1}{2}\), the angle \(\theta\) is \(60^\circ\). Understanding this relationship can assist in various applications, such as solving optimization problems in physics and engineering.

Determining the angle aids in analyzing the correlations between vectors, which is valuable for modeling and vector field analysis.

Dot Product

The dot product, also known as the scalar product, is a way to multiply two vectors to get a scalar (a single number). This operation is fundamental in physics and engineering because it provides a measure of two vectors' alignment or direction similarity. If the dot product is positive, the vectors point in generally the same direction; if negative, they point in opposite directions.The dot product formula for vectors \(\vec{b}\) and \(\vec{c}\) is given by:\[\vec{b} \cdot \vec{c} = |\vec{b}| |\vec{c}| \cos \theta\]Where:

• \(|\vec{b}|\) and \(|\vec{c}|\) are magnitudes of vectors \(\vec{b}\) and \(\vec{c}\).
• \(\theta\) is the angle between vectors.

The dot product's value can also be obtained by using the formula:\[\vec{b} \cdot \vec{c} = b_1c_1 + b_2c_2 + b_3c_3\]where clearly identifying each component product and adding them up gives the total dot product. This double utilization provides versatility in how you can approach vector problems.

Magnitude of Vectors

The magnitude of a vector represents its length or size, essentially a scalar quantity describing how "long" the vector is. It is calculated as the square root of the sum of the squares of its components in a space. Understanding and calculating magnitudes is crucial because they provide essential insights into the scale of physical quantities that vectors represent, such as displacement or force.To find the magnitude of a vector \(\vec{v} = (v_1, v_2, v_3)\), use the formula:\[|\vec{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}\]Here are some key points about magnitudes:

• Magnitudes are always non-negative.
• They give the "length" of the vector in the spatial context.
• The triangle inequality applies: \(\|\vec{u} + \vec{v}\| \leq \|\vec{u}\| + \|\vec{v}\|\).

In practical applications, knowing a vector's magnitude helps set up equations and solve for unknowns, especially when combined with other vectors' magnitudes and directions. It's a foundational concept that enhances your analytical skills when dealing with problems in multi-dimensional spaces.