Problem 97 - A Complete Resource Book in Mathematics for JEE Main 2017

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 [2008] (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 radians, option (A).

Step 1: Understand the Given Relationships

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

Step 2: Determine the Direction

Since \(\vec{a}\) and \(\vec{c}\) are scalar multiples of \(\vec{b}\), it means \(\vec{a}\) and \(\vec{c}\) point in the same direction. This is because multiplying a vector by a positive scalar does not change its direction.

Step 3: Use Collinearity to Find Angle

If two vectors are collinear and have the same direction, the angle between them is \(0\) radians (or \(0^\circ\)), since they are perfectly aligned.

Key Concepts

Scalar Multiplication

Scalar multiplication in vector algebra is the operation of multiplying a vector by a scalar, which is a single number. When you perform scalar multiplication, it affects two main things:

• The magnitude of the vector
• The direction remains unchanged if the scalar is positive

For example, if you have a vector \(\vec{b}\) and you multiply it by the scalar value 8, it means the vector \(\vec{a} = 8 \vec{b}\) stretches \(\vec{b}\) to be 8 times longer than its original length. Similarly, multiplying \(\vec{b}\) by 7 gives \(\vec{c} = 7 \vec{b}\), making \(\vec{c}\) 7 times the length of \(\vec{b}\).
Scalar multiplication is straightforward because:

• If the scalar is positive, the direction of the resultant vector remains the same.
• If the scalar is negative, the direction is reversed.

Using scalar multiplication helps us easily manipulate and scale vectors according to different requirements, particularly in the case of vectors like \(\vec{a}\) and \(\vec{c}\), which are based on a reference vector \(\vec{b}\).

Vector Direction

The direction of a vector is crucial in understanding how vectors like \(\vec{a}\) and \(\vec{c}\) relate to each other. Direction can be thought of as the 'angle' or 'orientation' in which the vector is pointing. When a vector is multiplied by a positive scalar, its direction remains unchanged.
Consider our vectors, \(\vec{a} = 8 \vec{b}\) and \(\vec{c} = 7 \vec{b}\). Both are positive scalar multiples of \(\vec{b}\), meaning they maintain the same direction as \(\vec{b}\). This implies they all point along the same line.
When vectors maintain the same direction:

• They are considered to be parallel.
• The angle between them is \(0\) radians if they are pointing in exactly the same direction.

Being able to determine vector direction is key in determining angles and relationships between different vectors such as in problems of vector algebra.

Collinearity

Collinearity in the context of vectors refers to vectors that lie along the same straight line. In the exercise, we see that \(\vec{a} = 8 \vec{b}\) and \(\vec{c} = 7 \vec{b}\), suggesting that all three vectors are multiples of \(\vec{b}\) and therefore collinear.
Collinearity has several implications:

• Vectors can be deemed collinear if they can be expressed as scalar multiples of another vector.
• The vectors will have the same or exact opposite direction.
• The angle between collinear vectors, when facing the same way, will always be \(0\) radians, meaning there is no angular separation.

In practice, whenever vectors are collinear, calculations simplify as you don't have to account for angular shifts. This makes it easier to analyze their sum, difference, and any dot product computations you might need to perform. In this case, because \(\vec{a}\) and \(\vec{c}\) are collinear with \(\vec{b}\), the angle between \(\vec{a}\) and \(\vec{c}\) is \(0\) radians, indicating perfect alignment.