Question:
If \(\vec{a}+\vec{b}+\vec{c}=0\) and \(|\vec{a}|=7,\;|\vec{b}|=5,\;|\vec{c}|=3\), then the angle between \(\vec{b}\) and \(\vec{c}\) is
If \(\vec{a}+\vec{b}+\vec{c}=0\) and \(|\vec{a}|=7,\;|\vec{b}|=5,\;|\vec{c}|=3\), then the angle between \(\vec{b}\) and \(\vec{c}\) is
Show Hint
If \(\vec{a}+\vec{b}+\vec{c}=0\), then one vector can be expressed as the negative of the sum of the other two vectors, and the magnitude formula can be used to find the angle between them.
Updated On: Jun 22, 2026
- \(30^\circ\)
- \(45^\circ\)
- \(60^\circ\)
- \(90^\circ\)
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The Correct Option is C
Solution and Explanation
Step 1: Use the given vector equation.
Given,
\[ \vec{a}+\vec{b}+\vec{c}=0 \] Therefore,
\[ \vec{a}=-(\vec{b}+\vec{c}) \]
Step 2: Take magnitude on both sides.
Since magnitude is always positive,
\[ |\vec{a}|=|\vec{b}+\vec{c}| \] Squaring both sides,
\[ |\vec{a}|^2=|\vec{b}+\vec{c}|^2 \]
Step 3: Apply the formula for magnitude of vector sum.
Let the angle between \(\vec{b}\) and \(\vec{c}\) be \(\theta\).
\[ |\vec{b}+\vec{c}|^2=|\vec{b}|^2+|\vec{c}|^2+2|\vec{b}||\vec{c}|\cos\theta \] Substituting the given values,
\[ 7^2=5^2+3^2+2(5)(3)\cos\theta \] \[ 49=25+9+30\cos\theta \] \[ 49=34+30\cos\theta \]
Step 4: Find the value of \(\cos\theta\).
\[ 30\cos\theta=49-34 \] \[ 30\cos\theta=15 \] \[ \cos\theta=\frac{15}{30} \] \[ \cos\theta=\frac{1}{2} \]
Step 5: Find the angle.
Since,
\[ \cos 60^\circ=\frac{1}{2} \] Therefore,
\[ \theta=60^\circ \]
Step 6: Final conclusion.
Hence, the angle between \(\vec{b}\) and \(\vec{c}\) is
\[ \boxed{60^\circ} \]
Given,
\[ \vec{a}+\vec{b}+\vec{c}=0 \] Therefore,
\[ \vec{a}=-(\vec{b}+\vec{c}) \]
Step 2: Take magnitude on both sides.
Since magnitude is always positive,
\[ |\vec{a}|=|\vec{b}+\vec{c}| \] Squaring both sides,
\[ |\vec{a}|^2=|\vec{b}+\vec{c}|^2 \]
Step 3: Apply the formula for magnitude of vector sum.
Let the angle between \(\vec{b}\) and \(\vec{c}\) be \(\theta\).
\[ |\vec{b}+\vec{c}|^2=|\vec{b}|^2+|\vec{c}|^2+2|\vec{b}||\vec{c}|\cos\theta \] Substituting the given values,
\[ 7^2=5^2+3^2+2(5)(3)\cos\theta \] \[ 49=25+9+30\cos\theta \] \[ 49=34+30\cos\theta \]
Step 4: Find the value of \(\cos\theta\).
\[ 30\cos\theta=49-34 \] \[ 30\cos\theta=15 \] \[ \cos\theta=\frac{15}{30} \] \[ \cos\theta=\frac{1}{2} \]
Step 5: Find the angle.
Since,
\[ \cos 60^\circ=\frac{1}{2} \] Therefore,
\[ \theta=60^\circ \]
Step 6: Final conclusion.
Hence, the angle between \(\vec{b}\) and \(\vec{c}\) is
\[ \boxed{60^\circ} \]
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