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

Given:

• Vector \( \overrightarrow{P} = |\overrightarrow{P}| \hat{i} \)
• Magnitude of vector \( \overrightarrow{Q} \), \( |\overrightarrow{Q}| = 10 \)
• Resultant vector \( \overrightarrow{R} = 3|\overrightarrow{P}| \hat{j} \)

Step 1: Express Vector Addition

The vector addition \( \overrightarrow{P} + \overrightarrow{Q} = \overrightarrow{R} \) can be written as:

\[ |\overrightarrow{P}| \hat{i} + \overrightarrow{Q} = 3|\overrightarrow{P}| \hat{j} \]

Step 2: Determine Components of \( \overrightarrow{Q} \)

Let \( \overrightarrow{Q} = Q_x \hat{i} + Q_y \hat{j} \). Then:

\[ |\overrightarrow{P}| \hat{i} + Q_x \hat{i} + Q_y \hat{j} = 3|\overrightarrow{P}| \hat{j} \]

Equate the components:

• For \( \hat{i} \) component: \( |\overrightarrow{P}| + Q_x = 0 \) ⇒ \( Q_x = -|\overrightarrow{P}| \)
• For \( \hat{j} \) component: \( Q_y = 3|\overrightarrow{P}| \)

Step 3: Use Magnitude of \( \overrightarrow{Q} \)

Given \( |\overrightarrow{Q}| = 10 \), we have:

\[ \sqrt{Q_x^2 + Q_y^2} = 10 \]

Substitute \( Q_x \) and \( Q_y \):

\[ \sqrt{(-|\overrightarrow{P}|)^2 + (3|\overrightarrow{P}|)^2} = 10 \]

\[ \sqrt{|\overrightarrow{P}|^2 + 9|\overrightarrow{P}|^2} = 10 \]

\[ \sqrt{10|\overrightarrow{P}|^2} = 10 \]

\[ |\overrightarrow{P}| \sqrt{10} = 10 \]

\[ |\overrightarrow{P}| = \frac{10}{\sqrt{10}} = \sqrt{10} \]

Conclusion:

The magnitude of \( \overrightarrow{P} \) is \( \sqrt{10} \).

Answer: \(\boxed{A}\)

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): \(\sqrt{10}\)

Answer 3

Step 1: Analyze the given vector equation.

We are given the vector addition equation:

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

The vectors are described as:

• \(\vec{P} = |\vec{P}| \hat{i}\), meaning \(\vec{P}\) lies entirely along the \(x\)-axis with magnitude \(|\vec{P}|\).
• \(|\vec{Q}| = 10\), meaning the magnitude of \(\vec{Q}\) is 10, but its direction is unspecified.
• \(\vec{R} = 3|\vec{P}| \hat{j}\), meaning \(\vec{R}\) lies entirely along the \(y\)-axis with magnitude \(3|\vec{P}|\).

 

Step 2: Break the vectors into components.

From the given information:

• \(\vec{P} = |\vec{P}| \hat{i}\), so the \(x\)-component of \(\vec{P}\) is \(|\vec{P}|\) and the \(y\)-component is 0.
• \(\vec{R} = 3|\vec{P}| \hat{j}\), so the \(x\)-component of \(\vec{R}\) is 0 and the \(y\)-component is \(3|\vec{P}|\).

 

Let \(\vec{Q} = Q_x \hat{i} + Q_y \hat{j}\). Using the vector addition equation, we can write: \[ \vec{P} + \vec{Q} = \vec{R}. \]

In component form: \[ (|\vec{P}| + Q_x) \hat{i} + Q_y \hat{j} = 0 \hat{i} + 3|\vec{P}| \hat{j}. \]

Equating the \(x\)- and \(y\)-components: \[ |\vec{P}| + Q_x = 0 \quad \text{(1)}, \] \[ Q_y = 3|\vec{P}| \quad \text{(2)}. \]

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Step 3: Use the magnitude of \(\vec{Q}\).

The magnitude of \(\vec{Q}\) is given as \(|\vec{Q}| = 10\). The magnitude of a vector is related to its components by: \[ |\vec{Q}| = \sqrt{Q_x^2 + Q_y^2}. \]

Substitute \(Q_x\) from equation (1): \[ Q_x = -|\vec{P}|. \] Substitute \(Q_y\) from equation (2): \[ Q_y = 3|\vec{P}|. \] Thus: \[ |\vec{Q}| = \sqrt{(-|\vec{P}|)^2 + (3|\vec{P}|)^2}. \] Simplify: \[ 10 = \sqrt{|\vec{P}|^2 + 9|\vec{P}|^2}. \] \[ 10 = \sqrt{10|\vec{P}|^2}. \] Square both sides: \[ 100 = 10|\vec{P}|^2. \] Solve for \(|\vec{P}|^2\): \[ |\vec{P}|^2 = 10. \] Take the square root: \[ |\vec{P}| = \sqrt{10}. \] ---

Final Answer: The magnitude of \(\vec{P}\) is \( \mathbf{\sqrt{10}} \), which corresponds to option \( \mathbf{(A)} \).