Question:
Let the position vectors of points \(A, B, C\) be \( \vec{a}, \vec{b}, \vec{c} \) respectively. Let \(Q\) be the centroid. Then \( \overrightarrow{QA} + \overrightarrow{QB} + \overrightarrow{QC} = \)
Let the position vectors of points \(A, B, C\) be \( \vec{a}, \vec{b}, \vec{c} \) respectively. Let \(Q\) be the centroid. Then \( \overrightarrow{QA} + \overrightarrow{QB} + \overrightarrow{QC} = \)
Show Hint
Sum of vectors from centroid to vertices of triangle is always zero.
Updated On: May 8, 2026
- \( \frac{\vec{a}+\vec{b}+\vec{c}}{2} \)
- \( 2\vec{a}+\vec{b}+\vec{c} \)
- \( \vec{a}+\vec{b}+\vec{c} \)
- \( \frac{\vec{a}+\vec{b}+\vec{c}}{3} \)
- \( \vec{0} \)
Show Solution
Verified By Collegedunia
The Correct Option is
Solution and Explanation
Concept:
• Centroid: \[ \vec{Q} = \frac{\vec{a}+\vec{b}+\vec{c}}{3} \]
Step 1: Write vectors.
\[ \overrightarrow{QA} = \vec{a} - \vec{Q} \] \[ \overrightarrow{QB} = \vec{b} - \vec{Q} \] \[ \overrightarrow{QC} = \vec{c} - \vec{Q} \]
Step 2: Add them.
\[ (\vec{a} - \vec{Q}) + (\vec{b} - \vec{Q}) + (\vec{c} - \vec{Q}) \] \[ = \vec{a} + \vec{b} + \vec{c} - 3\vec{Q} \]
Step 3: Substitute centroid.
\[ 3\vec{Q} = \vec{a} + \vec{b} + \vec{c} \]
Step 4: Simplify.
\[ = 0 \]
Step 5: Final Answer.
\[ \boxed{\vec{0}} \]
• Centroid: \[ \vec{Q} = \frac{\vec{a}+\vec{b}+\vec{c}}{3} \]
Step 1: Write vectors.
\[ \overrightarrow{QA} = \vec{a} - \vec{Q} \] \[ \overrightarrow{QB} = \vec{b} - \vec{Q} \] \[ \overrightarrow{QC} = \vec{c} - \vec{Q} \]
Step 2: Add them.
\[ (\vec{a} - \vec{Q}) + (\vec{b} - \vec{Q}) + (\vec{c} - \vec{Q}) \] \[ = \vec{a} + \vec{b} + \vec{c} - 3\vec{Q} \]
Step 3: Substitute centroid.
\[ 3\vec{Q} = \vec{a} + \vec{b} + \vec{c} \]
Step 4: Simplify.
\[ = 0 \]
Step 5: Final Answer.
\[ \boxed{\vec{0}} \]
Was this answer helpful?
0
0
Top KEAM Mathematics Questions
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
- The value of $ \cos [{{\tan }^{-1}}\{\sin ({{\cot }^{-1}}x)\}] $ is
- The solutions set of inequation $\cos^{-1}x < \,\sin^{-1}x$ is
- Let $\Delta= \begin{vmatrix}1&1&1\\ 1&-1-w^{2}&w^{2}\\ 1&w&w^{4}\end{vmatrix}$, where $w \neq 1$ is a complex number such that $w^3 = 1$. Then $\Delta$ equals
- Let $p : 57$ is an odd prime number, $\quad \, q : 4$ is a divisor of $12$ $\quad$ $r : 15$ is the $LCM$ of $3$ and $5$ Be three simple logical statements. Which one of the following is true?
View More Questions
Top KEAM Vector basics Questions
The projection of the vector \(a=2i-3j+4k \)on the vector \(b=i+2j+2k\) is ?
- If $2\hat{i} - \hat{j} + \hat{k} = s(3\hat{i} - 4\hat{j} - 4\hat{k}) + t(\hat{i} - 3\hat{j} - 5\hat{k})$, where $s$ and $t$ are scalars, then $3s + 5t$ is equal to:
- The vectors $2\vec{a}+6\vec{b}$ and $3\vec{a}-7\vec{b}$ are position vectors of the points A and B respectively. A point P divides the line segment AB internally in the ratio 3:5. Then $\vec{PB}=$
- The position vectors of the vertices of a triangle are $\hat{i}+2\hat{j}-4\hat{k}$, $-\hat{i}-2\hat{j}-4\hat{k}$ and $2\hat{i}+3\hat{j}+5\hat{k}$. The perimeter of the triangle is
- Let $|\vec{a}|=2,|\vec{b}|=\sqrt{13+6\sqrt{3}}$ and $|\vec{c}|=3$. If $\vec{a}-\vec{b}+\vec{c}=0$, then the angle between $\vec{a}$ and $\vec{c}$ is
View More Questions
Top KEAM Questions
- i.$\quad$ They help in respiration ii.$\quad$ They help in cell wall formation iii.$\quad$ They help in DNA replication iv. $\quad$They increase surface area of plasma membrane Which of the following prokaryotic structures has all the above roles?
- A body oscillates with SHM according to the equation (in SI units), $x = 5 cos \left(2\pi t +\frac{\pi}{4}\right) .$ Its instantaneous displacement at $t = 1$ second is
- The pH of a solution obtained by mixing 60 mL of 0.1 M BaOH solution at 40m of 0.15m HCI solution is
Kepler's second law (law of areas) of planetary motion leads to law of conservation of
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
View More Questions