Problem 97 - A Complete Resource Book in Mathematics for JEE Main

Question

The non-zero vectors \(\vec{a}, \vec{b}\) and \(\vec{c}\) are related by \(\vec{a}=8 \vec{b}\) and \(\vec{c}=7 \vec{b}\). Then the angle between \(\vec{a}\) and \(\vec{c}\) is \([\mathbf{2 0 0 8}]\) (A) 0 (B) \(\pi / 4\) (C) \(\pi / 2\) (D) \(\pi\)

Step-by-Step Solution

Answer

The angle between \(\vec{a}\) and \(\vec{c}\) is 0.

Step 1: Understand the Relationship between Vectors

We are given that \(\vec{a} = 8 \vec{b}\) and \(\vec{c} = 7 \vec{b}\). This means that both vectors \(\vec{a}\) and \(\vec{c}\) are scalar multiples of the same vector \(\vec{b}\).

Step 2: Express Vectors in Terms of Each Other

Since \(\vec{a} = 8 \vec{b}\) and \(\vec{c} = 7 \vec{b}\), the vectors \(\vec{a}\) and \(\vec{c}\) can be rewritten as direct scalar multiples of \(\vec{b}\): \(\vec{a} = 8 \cdot \vec{b},\) and \(\vec{c} = 7 \cdot \vec{b}\). This shows both are in the direction of \(\vec{b}\).

Step 3: Determine the Angle Between the Vectors

The angle between any two vectors that are scalar multiples of the same vector is either \(0\) if they are in the same direction, or \(\pi\) if they are in opposite directions. Since 8 and 7 are both positive scalars, \(\vec{a}\) and \(\vec{c}\) are in the same direction. Hence, the angle between them is \(0\).

Key Concepts

Scalar Multiples

Understanding scalar multiples is crucial in vector mathematics. A vector is called a scalar multiple of another if it can be expressed as the product of a scalar (a real number) with another vector. For instance, if \(\vec{v}\) is a vector, then \(k\vec{v}\) is a vector that points in the same direction as \(\vec{v}\) (or directly opposite if \(k\) is negative), but its magnitude is \(|k|\) times the magnitude of \(\vec{v}\).

In the exercise, vectors \(\vec{a}\) and \(\vec{c}\) are scalar multiples of \(\vec{b}\), as \(\vec{a} = 8 \vec{b}\) and \(\vec{c} = 7 \vec{b}\). This relationship illustrates that both vectors \(\vec{a}\) and \(\vec{c}\) maintain the same direction as \(\vec{b}\).

This property simplifies calculations, especially when determining the angle between such vectors. Recognizing scalar multiples helps in understanding many problems in physics and geometry, where vectors represent various quantities like forces or velocities.

Angle Between Vectors

The angle between two vectors is a measure of their directional difference. To calculate this angle \(\theta\), we use the dot product formula:

\[\vec{a} \cdot \vec{c} = ||\vec{a}|| ||\vec{c}|| \cos(\theta)\]

This equation allows us to solve for \(\cos(\theta)\) and subsequently find \(\theta\).

However, when vectors are scalar multiples of the same vector, as in the exercise, the calculation is simpler. If the scalar multiples are positive, the vectors point in the same direction and the angle between them is \(0\). If one scalar is positive and the other is negative, the vectors point in opposite directions, making the angle \(\pi\).

Understanding the angle between vectors is fundamental in physics for analyzing forces, movement, and more, as it tells us how much one direction differs from another.

Direction of Vectors

The direction of a vector describes its orientation in space. Vectors are often visualized as arrows where the length represents magnitude and the arrowhead shows direction.

In mathematical terms, the direction is given by the vector's components along the coordinate axes. Two vectors have the same direction when they are scalar multiples of each other.

In the exercise, the vectors \(\vec{a}\) and \(\vec{c}\) both align with the vector \(\vec{b}\), showing that they have the same direction, as they are both expressed as \(\vec{a} = 8 \vec{b}\) and \(\vec{c} = 7 \vec{b}\).

This concept is essential for understanding vector addition, subtraction, and applications in navigation where the precise path or trajectory is critical.