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:
We are given the coordinates of point \(P\) and the vector \(\vec{PQ}\). We can find the position vector of point \(Q\) by adding the position vector of \(P\) to \(\vec{PQ}\). Finally, we calculate the distance of \(Q\) from the origin.
Step 2: Key Formula or Approach:
The position vector of \(Q\) is \(\vec{OQ} = \vec{OP} + \vec{PQ}\).
The distance of \(Q(x, y, z)\) from the origin is given by \(|\vec{OQ}| = \sqrt{x^2 + y^2 + z^2}\).
Step 3: Detailed Explanation:
The position vector of \(P\) is \(\vec{OP} = 2\hat{i} + \hat{j} - 3\hat{k}\).
We are given \(\vec{PQ} = 3\hat{i} - \hat{j} + 5\hat{k}\).
Using vector addition:
\[ \vec{OQ} = \vec{OP} + \vec{PQ} \] \[ \vec{OQ} = (2\hat{i} + \hat{j} - 3\hat{k}) + (3\hat{i} - \hat{j} + 5\hat{k}) \] \[ \vec{OQ} = (2 + 3)\hat{i} + (1 - 1)\hat{j} + (-3 + 5)\hat{k} \] \[ \vec{OQ} = 5\hat{i} + 0\hat{j} + 2\hat{k} \] The coordinates of \(Q\) are \((5, 0, 2)\).
The distance of \(Q\) from the origin \(O\) is the magnitude of \(\vec{OQ}\):
\[ |\vec{OQ}| = \sqrt{5^2 + 0^2 + 2^2} = \sqrt{25 + 0 + 4} = \sqrt{29} \] Step 4: Final Answer:
The distance is \(\sqrt{29}\).