Problem 49 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The non-zero vectors are \(\vec{a}, \vec{b}\) 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) \(\frac{\pi}{4}\) (c) \(\frac{\pi}{2}\) (d) \(\pi\)

Step-by-Step Solution

Answer

The angle is \(\pi\), so the answer is (d).

Step 1: Understand the Problem

We need to find the angle between the vectors \(\vec{a}\) and \(\vec{c}\). The vectors are related by \(\vec{a}=8 \vec{b}\) and \(\vec{c}=-7 \vec{b}\).

Step 2: Determine the Relationships between Vectors

Given \(\vec{a} = 8 \vec{b}\) and \(\vec{c} = -7 \vec{b}\), we can see that \(\vec{a}\) and \(\vec{c}\) are scalar multiples of \(\vec{b}\). Thus, \(\vec{a}\) and \(\vec{c}\) are collinear but point in opposite directions because of the negative sign in \(\vec{c} = -7 \vec{b}\).

Step 3: Determine the Angle Between Oppositely Directed Vectors

When two non-zero vectors are oppositely directed, the angle between them is \(\pi\) radians because they point in exactly opposite directions.

Step 4: Conclude the Solution

Therefore, the angle between \(\vec{a}\) and \(\vec{c}\) is \(\pi\) radians, which corresponds to option (d) in the list of possible answers.

Key Concepts

angle between vectors

The angle between vectors is a fundamental concept in vector algebra that helps us understand how vectors relate to each other in space. It determines how much one vector needs to rotate to align with another. When two vectors point in the same direction, their angle is 0 degrees (or 0 radians). But if they point in exactly opposite directions, like \(\vec{a}\) and \(\vec{c}\) in our exercise, the angle is \(\pi\) radians, which is equivalent to 180 degrees.
It's important to note that the angle between vectors is commonly found using the dot product formula:

• \(\vec{a} \cdot \vec{c} = ||\vec{a}|| ||\vec{c}|| \cos(\theta)\)
• \(\theta\) is the angle between vectors \(\vec{a}\) and \(\vec{c}\)

However, in this case, because the vectors are scalar multiples, they are collinear and simply vector geometry tells us they are in opposite directions.

scalar multiplication of vectors

Scalar multiplication involves multiplying a vector by a scalar (a real number). When a vector \(\vec{b}\) is multiplied by a scalar, each component of the vector is multiplied by that scalar. For instance, multiplying \(\vec{b}\) by 8 to obtain \(\vec{a} = 8\vec{b}\) increases the magnitude of \(\vec{b}\) eight times, but does not change its direction.
In contrast, multiplying \(\vec{b}\) by -7 to form \(\vec{c} = -7\vec{b}\) changes both the magnitude and direction of the original vector \(\vec{b}\). The negative sign indicates that the resulting vector \(\vec{c}\) points in the opposite direction of \(\vec{b}\).

• Multiplying by a positive scalar: keeps direction, changes magnitude.
• Multiplying by a negative scalar: reverses direction, changes magnitude.

This concept is crucial in understanding how vectors relate, particularly in determining if they're collinear or not.

collinear vectors

Collinear vectors are vectors that lie along the same line. In other words, they have the same or exact opposite direction. If two vectors are collinear, you can express one as a scalar multiple of the other. For example, vectors \(\vec{a} = 8\vec{b}\) and \(\vec{c} = -7\vec{b}\) lie along the same line due to being scalar multiples of the same vector \(\vec{b}\).
Even though \(\vec{a}\) and \(\vec{c}\) are not the same length, their directions are either parallel or directly opposite.
Why is this important? Understanding that vectors are collinear helps in simplifying many vector problems because:

• Any linear combination or comparison with other vectors becomes straightforward.
• It is easier to predict movements or interactions in physics when vectors are on the same line.

Being collinear dictates that the angle between them can only be 0 or \(\pi\) radians, depending on whether they point in the same or opposite directions respectively.