Question

Let \( \vec{a} \) and \( \vec{b} \) be two vectors such that \( |\vec{a}| = |\vec{b}| \) and \( |\vec{a}+2\vec{b}| = |2\vec{a}-\vec{b}| \). If \( \vec{c} \) is a vector parallel to \( \vec{a} \) then the angle between \( \vec{b} \) and \( \vec{c} \) is

• A. \( 0^\circ \)
• B. \( 30^\circ \)
• C. \( 60^\circ \)
• D. \( 90^\circ \)

Hint:

When an equation involves magnitudes of vector sums or differences, squaring both sides is a powerful technique. It converts the problem into dot products using the identity \( |\vec{v}|^2 = \vec{v} \cdot \vec{v} \), which often leads to a simple relationship between the vectors.

Answer 1

The Correct Option is D

We are given the condition \( |\vec{a}+2\vec{b}| = |2\vec{a}-\vec{b}| \).
To work with this, we can square both sides of the equation.
\( |\vec{a}+2\vec{b}|^2 = |2\vec{a}-\vec{b}|^2 \).
Using the property \( |\vec{v}|^2 = \vec{v} \cdot \vec{v} \), we expand both sides.
\( (\vec{a}+2\vec{b}) \cdot (\vec{a}+2\vec{b}) = (2\vec{a}-\vec{b}) \cdot (2\vec{a}-\vec{b}) \).
\( \vec{a}\cdot\vec{a} + 2(\vec{a}\cdot\vec{b}) + 2(\vec{b}\cdot\vec{a}) + 4(\vec{b}\cdot\vec{b}) = 4(\vec{a}\cdot\vec{a}) - 2(\vec{a}\cdot\vec{b}) - 2(\vec{b}\cdot\vec{a}) + \vec{b}\cdot\vec{b} \).
This simplifies to \( |\vec{a}|^2 + 4(\vec{a}\cdot\vec{b}) + 4|\vec{b}|^2 = 4|\vec{a}|^2 - 4(\vec{a}\cdot\vec{b}) + |\vec{b}|^2 \).
We are also given that \( |\vec{a}| = |\vec{b}| \). Let's substitute \( |\vec{b}| \) with \( |\vec{a}| \).
\( |\vec{a}|^2 + 4(\vec{a}\cdot\vec{b}) + 4|\vec{a}|^2 = 4|\vec{a}|^2 - 4(\vec{a}\cdot\vec{b}) + |\vec{a}|^2 \).
\( 5|\vec{a}|^2 + 4(\vec{a}\cdot\vec{b}) = 5|\vec{a}|^2 - 4(\vec{a}\cdot\vec{b}) \).
Subtract \( 5|\vec{a}|^2 \) from both sides and add \( 4(\vec{a}\cdot\vec{b}) \) to both sides.
\( 8(\vec{a}\cdot\vec{b}) = 0 \).
This implies \( \vec{a}\cdot\vec{b} = 0 \).
The dot product of two non-zero vectors is zero only if they are perpendicular to each other. So, the angle between \( \vec{a} \) and \( \vec{b} \) is \( 90^\circ \).
We are told that vector \( \vec{c} \) is parallel to vector \( \vec{a} \). This means \( \vec{c} \) has the same direction as \( \vec{a} \).
Therefore, the angle between \( \vec{b} \) and \( \vec{c} \) is the same as the angle between \( \vec{b} \) and \( \vec{a} \), which is \( 90^\circ \).