Question

Consider a vector addition \(\vec{P}\)+\(\vec{Q}\)=\(\vec{R}\). If \(\vec{P}\)=|\(\vec{P}\)|\(\hat{i}\),|\(\vec{Q}\)|=10 and\(\vec{R}\)= 3 |\(\vec{P}\)|\(\hat{j}\),then |\(\vec{P}\)| is:

• A. \(\sqrt{10}\)
• B. 30
• C. \(\sqrt{30}\)
• D. 2\(\sqrt{10}\)
• E. 2\(\sqrt{20}\)

Answer 1

The Correct Option is A

The correct answer is (A): \(\sqrt{10}\)

Answer 2

Let's analyze the given vector addition problem:

\[\vec{P} + \vec{Q} = \vec{R}\]

We are given:
- \(\vec{P} = |\vec{P}| \hat{i}\)
- \(|\vec{Q}| = 10\)
- \(\vec{R} = 3 |\vec{P}| \hat{j}\)

Let's denote \(|\vec{P}|\) as \(P\). Therefore, \(\vec{P} = P \hat{i}\).

Given the magnitude and direction of \(\vec{Q}\) are not directly specified, we need to find out its components.

The vector equation can be written in component form:
\[P \hat{i} + \vec{Q} = 3P \hat{j}\]

Let \(\vec{Q} = Q_x \hat{i} + Q_y \hat{j}\).

Thus,
\[P \hat{i} + Q_x \hat{i} + Q_y \hat{j} = 0 \hat{i} + 3P \hat{j}\]

From the above equation, we can separate it into its \(i\)-component and \(j\)-component equations:
\[P + Q_x = 0\]
\[Q_y = 3P\]

From the first equation:
\[Q_x = -P\]

Given \(|\vec{Q}| = 10\):
\[|\vec{Q}| = \sqrt{Q_x^2 + Q_y^2}\]

Substitute \(Q_x\) and \(Q_y\):
\[10 = \sqrt{(-P)^2 + (3P)^2}\]
\[10 = \sqrt{P^2 + 9P^2}\]
\[10 = \sqrt{10P^2}\]
\[10 = \sqrt{10} \cdot P\]
\[10 = \sqrt{10} P\]
\[P = \frac{10}{\sqrt{10}} = \sqrt{10}\]

Thus, the magnitude of \(\vec{P}\), \(|\vec{P}|\), is \(\sqrt{10}\).
So The correct answer is option (A): \(\sqrt{10}\)