Let the two vectors be \( \vec{A} \) and \( \vec{B} \) with magnitudes \( A \) and \( B \), respectively. Then, the magnitude of their sum is given by:
\(|\overrightarrow{ A } + \overrightarrow{ B }| = \sqrt{A^2 + B^2 + 2AB \cos \theta} \) (\)
Where \( \theta \) is the angle between the vectors.
The magnitude of their difference is given by:
\(|\overrightarrow{ A } - \overrightarrow{ B }| = \sqrt{A^2 + B^2 - 2AB \cos \theta} \) (\)
Now, according to the question, we are given that:
\(|\overrightarrow{ A } + \overrightarrow{ B }| = |\overrightarrow{ A } - \overrightarrow{ B }|\)
From this, we have:
\(A^2 + B^2 + 2AB \cos \theta = A^2 + B^2 - 2AB \cos \theta\)
Simplifying the above equation:
\(4AB \cos \theta = 0\)
This gives us:
\(\cos \theta = 0\)
Thus, the angle between the two vectors is:
\(\theta = 90^\circ\)
The angle between the vectors \( \vec{A} \) and \( \vec{B} \) is \( 90^\circ \), meaning the vectors are perpendicular to each other.
Two vectors \(\overrightarrow{A}+\overrightarrow{B}\) have equal magnitudes. If magnitude of \(\overrightarrow{A}+\overrightarrow{B}\) is equal to two times the magnitude of \(\overrightarrow{A}-\overrightarrow{B}\) then the angle between vec A and \(\overrightarrow{B}\) will be
List I | List II | ||
---|---|---|---|
A | Mesozoic Era | I | Lower invertebrates |
B | Proterozoic Era | II | Fish & Amphibia |
C | Cenozoic Era | III | Birds & Reptiles |
D | Paleozoic Era | IV | Mammals |
Vector Quantity is a physical quantity that is specified not only by its magnitude but also by its direction. A vector quantity whose magnitude is equal to one and has direction is called a unit vector.
Examples of vector quantity are-