Question

Let $P(2, 1, -3)$ be a given point. If the point $Q$ is such that $\vec{PQ} = 3\hat{i} - \hat{j} + 5\hat{k}$, then the distance of $Q$ from the origin $O$ is

• A. $\sqrt{29}$
• B. $\sqrt{33}$
• C. $\sqrt{59}$
• D. $\sqrt{69}$
• E. $\sqrt{79}$

Hint:

In 3D geometry, adding a vector to a starting point gives you the coordinates of the ending point. Just remember: $\text{End} = \text{Start} + \text{Vector}$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
To find the position of point $Q$, we use the vector relation $\vec{PQ} = \vec{OQ} - \vec{OP}$. Once we have the coordinates of $Q$, we can calculate its distance from the origin using the standard distance formula \( \sqrt{x^2 + y^2 + z^2} \).
Key Formula or Approach:
1. Find position vector of $Q$: $\vec{OQ} = \vec{OP} + \vec{PQ}$.
2. Distance from origin: $d = |\vec{OQ}|$.

Step 2: Detailed Explanation:
The position vector of point $P(2, 1, -3)$ is $\vec{OP} = 2\hat{i} + \hat{j} - 3\hat{k}$.
Given the displacement vector $\vec{PQ} = 3\hat{i} - \hat{j} + 5\hat{k}$.
Calculating $\vec{OQ}$:
\[ \vec{OQ} = (2\hat{i} + \hat{j} - 3\hat{k}) + (3\hat{i} - \hat{j} + 5\hat{k}) \]
\[ \vec{OQ} = (2+3)\hat{i} + (1-1)\hat{j} + (-3+5)\hat{k} \]
\[ \vec{OQ} = 5\hat{i} + 0\hat{j} + 2\hat{k} \]
Thus, the coordinates of $Q$ are $(5, 0, 2)$.
The distance of $Q$ from the origin $O(0, 0, 0)$ is:
\[ |OQ| = \sqrt{5^2 + 0^2 + 2^2} = \sqrt{25 + 0 + 4} = \sqrt{29} \]

Step 3: Final Answer:
The distance is $\sqrt{29}$.