When working with vectors, remember that the dot product \( \vec{a} \cdot \vec{b} = 0 \) implies that the vectors \( \vec{a} \) and \( \vec{b} \) are perpendicular. This property is useful in geometry and physics problems involving right-angled vectors. Also, when two vectors are opposite to each other, the angle between them is \( 180^\circ \), as they point in completely opposite directions.
Given:
\[\vec{a} + \vec{b} + \vec{c} = \vec{0} \Rightarrow \vec{c} = -(\vec{a} + \vec{b})\]
The magnitude of \(\vec{c}\) is:
\[|\vec{c}|^2 = |\vec{a} + \vec{b}|^2\]
Expand using the vector magnitude formula:
\[|\vec{c}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2\vec{a} \cdot \vec{b}\]
Substitute \(|\vec{a}| = |\vec{b}| = 1\) and \(|\vec{c}| = 2\):
\[2^2 = 1 + 1 + 2(\vec{a} \cdot \vec{b})\]
Simplify:
\[4 = 2 + 2(\vec{a} \cdot \vec{b})\]
Solve for \(\vec{a} \cdot \vec{b}\):
\[2(\vec{a} \cdot \vec{b}) = 2 \Rightarrow \vec{a} \cdot \vec{b} = 0\]
This means \(\vec{a}\) and \(\vec{b}\) are perpendicular.
From the equation \(\vec{c} = -(\vec{a} + \vec{b})\):
\[|\vec{a} + \vec{b}| = \sqrt{|\vec{a}|^2 + |\vec{b}|^2} = \sqrt{1^2 + 1^2} = \sqrt{2}\]
Thus:
\[\vec{c} = -(\vec{a} + \vec{b})\]
and its direction is opposite to \(\vec{a} + \vec{b}\).
Since \(\vec{c}\) is opposite to \(\vec{a} + \vec{b}\), and \(\vec{b}\) contributes to \(\vec{c}\), the angle between \(\vec{b}\) and \(\vec{c}\) is
\[\theta = 180^\circ.\]
Thus:
\[180^\circ.\]
Given:
\(\vec{a} + \vec{b} + \vec{c} = \vec{0} \Rightarrow \vec{c} = -(\vec{a} + \vec{b})\)
Step 1: Find the magnitude of \( \vec{c} \):
The magnitude of \( \vec{c} \) is given by: \[ |\vec{c}|^2 = |\vec{a} + \vec{b}|^2 \] Expanding using the vector magnitude formula: \[ |\vec{c}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2 \vec{a} \cdot \vec{b} \]Step 2: Substitute the known values:
We are given that \( |\vec{a}| = |\vec{b}| = 1 \) and \( |\vec{c}| = 2 \). Substituting these values into the equation: \[ 2^2 = 1^2 + 1^2 + 2 (\vec{a} \cdot \vec{b}) \] Simplifying: \[ 4 = 2 + 2 (\vec{a} \cdot \vec{b}) \]Step 3: Solve for \( \vec{a} \cdot \vec{b} \):
Rearranging the equation: \[ 2 (\vec{a} \cdot \vec{b}) = 2 \Rightarrow \vec{a} \cdot \vec{b} = 0 \] This means that \( \vec{a} \) and \( \vec{b} \) are perpendicular to each other.Step 4: Analyze the geometry of the vectors:
Since \( \vec{a} \) and \( \vec{b} \) are perpendicular, their resultant \( \vec{a} + \vec{b} \) forms the diagonal of a square with side length 1.Step 5: Find the magnitude of \( \vec{a} + \vec{b} \):
The magnitude of \( \vec{a} + \vec{b} \) is: \[ |\vec{a} + \vec{b}| = \sqrt{|\vec{a}|^2 + |\vec{b}|^2} = \sqrt{1^2 + 1^2} = \sqrt{2} \]Step 6: Magnitude of \( \vec{c} \):
Since \( \vec{c} = -(\vec{a} + \vec{b}) \), its magnitude is: \[ |\vec{c}| = \sqrt{2} \] and its direction is opposite to \( \vec{a} + \vec{b} \).Step 7: Find the angle between \( \vec{b} \) and \( \vec{c} \):
Since \( \vec{c} \) is opposite to \( \vec{a} + \vec{b} \), and \( \vec{b} \) contributes to \( \vec{c} \), the angle between \( \vec{b} \) and \( \vec{c} \) is: \[ \theta = 180^\circ \]Conclusion:
Thus, the angle between \( \vec{b} \) and \( \vec{c} \) is: \[ 180^\circ. \]List-I | List-II | ||
A | Megaliths | (I) | Decipherment of Brahmi and Kharoshti |
B | James Princep | (II) | Emerged in first millennium BCE |
C | Piyadassi | (III) | Means pleasant to behold |
D | Epigraphy | (IV) | Study of inscriptions |