Question

$\vec{P} = x\hat{i} + 0.8\hat{j} + 0.6\hat{k}$ If magnitude of $\vec{P}$ is 2, value of x is

• A. $\sqrt{3}$
• B. 2
• C. $\sqrt{2}$
• D. 1

Hint:

Recognizing common Pythagorean triplets (like $0.6, 0.8, 1.0$) can greatly speed up calculations! Since $0.8^2 + 0.6^2 = 1^2$, the equation instantly simplifies to $x^2 + 1 = 4$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A vector in three-dimensional space is represented by its components along the x, y, and z axes.
The magnitude (or length) of this vector is found by taking the square root of the sum of the squares of its individual scalar components.
Step 2: Key Formula or Approach:
For a given vector $\vec{P} = P_x\hat{i} + P_y\hat{j} + P_z\hat{k}$, the magnitude $|\vec{P}|$ is given by the formula:
\[ |\vec{P}| = \sqrt{P_x^2 + P_y^2 + P_z^2} \] Step 3: Detailed Explanation:
From the given vector equation, the components are $P_x = x$, $P_y = 0.8$, and $P_z = 0.6$.
The problem states that the magnitude of the vector is exactly 2, so $|\vec{P}| = 2$.
Substitute these given values into the magnitude formula:
\[ 2 = \sqrt{x^2 + (0.8)^2 + (0.6)^2} \] Square both sides of the equation to remove the square root:
\[ 4 = x^2 + 0.64 + 0.36 \] Add the numerical values on the right side:
\[ 4 = x^2 + 1.00 \] Subtract 1 from both sides to isolate $x^2$:
\[ x^2 = 4 - 1 \] \[ x^2 = 3 \] Taking the square root gives the value of $x$:
\[ x = \sqrt{3} \] Step 4: Final Answer:
The value of $x$ is $\sqrt{3}$.