Question

If $\text{G}(\vec{\text{g}})$, $\text{H}(\vec{\text{h}})$ and $\text{P}(\vec{\text{p}})$ are respectively centroid, orthocenter and circumcentre of a triangle and $x\vec{\text{p}} + y\vec{\text{h}} + z\vec{\text{g}} = \vec{\text{0}}$, then $x, y, z$ are respectively

• A. $1, 1, -2$
• B. $1, 3, -4$
• C. $2, 1, -3$
• D. $2, 3, -5$

Hint:

Remember the alphabetical mnemonic ordered sequence for the Euler line: H $\rightarrowG\rightarrow$ O (or P for circumcenter), with the ratio 2 : 1. Keeping the word "HG Out" or "HGP" in mind helps you write down the section formula correctly every single time!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem gives three position vectors representing the centroid, orthocenter, and circumcenter of a triangle. We need to identify the scalar constants $x$, $y$, and $z$ that satisfy a given linear vector combination equation.

Step 2: Key Formula or Approach:
In any triangle, a fundamental geometric property states that the Euler line connects the orthocenter (H), centroid (G), and circumcenter (P) in a straight path. Furthermore, the centroid G always divides the line segment joining the orthocenter H to the circumcenter P internally in a strict $2 : 1$ ratio.
Using the section formula for vectors:
$$\vec{\text{g}} = \frac{1\vec{\text{h}} + 2\vec{\text{p}}}{1 + 2}$$

Step 3: Detailed Explanation:
Let's simplify the section formula relation:
$$\vec{\text{g}} = \frac{\vec{\text{h}} + 2\vec{\text{p}}}{3}$$ Multiply both sides of the equation by 3:
$$3\vec{\text{g}} = \vec{\text{h}} + 2\vec{\text{p}}$$ Rearrange all vectors to one side to set the equation equal to the zero vector ($\vec{\text{0}}$):
$$2\vec{\text{p}} + 1\vec{\text{h}} - 3\vec{\text{g}} = \vec{\text{0}}$$ Now, let's compare this derived equation directly to the given expression format:
$$x\vec{\text{p}} + y\vec{\text{h}} + z\vec{\text{g}} = \vec{\text{0}}$$ Matching the corresponding scalar coefficients gives:
$x = 2$
$y = 1$
$z = -3$
This sequence matches option (C).

Step 4: Final Answer:
The scalar values are $2, 1, -3$, which corresponds to option (C).