Question

If the magnitude of a vector \( \vec{p} \) is 25 units and its y-component is 7 units, then its x-component is

• A. 24 units
• B. 18 units
• C. 32 units
• D. 16 units

Hint:

For a vector \( \vec{p} = p_x\vec{i} + p_y\vec{j} + p_z\vec{k} \), its magnitude is \( |\vec{p}| = \sqrt{p_x^2 + p_y^2 + p_z^2} \). If the problem only discusses x and y components, or does not provide information about the z-component, it's often implied that \(p_z=0\) or the problem is in 2D. So, \( |\vec{p}|^2 = p_x^2 + p_y^2 \).

Answer 1

The Correct Option is A

Let the vector be \( \vec{p} = p_x \vec{i} + p_y \vec{j} \), assuming it's a 2D vector.
If it's 3D, \( \vec{p} = p_x \vec{i} + p_y \vec{j} + p_z \vec{k} \).
The magnitude of the vector is \( |\vec{p}| \).
Given \( |\vec{p}| = 25 \) units.
Given the y-component \( p_y = 7 \) units.
If the vector is 2-dimensional (lies in xy-plane), then \( |\vec{p}| = \sqrt{p_x^2 + p_y^2} \).
\[ 25 = \sqrt{p_x^2 + 7^2} \] Square both sides: \[ 25^2 = p_x^2 + 7^2 \] \[ 625 = p_x^2 + 49 \] \[ p_x^2 = 625 - 49 = 576 \] \[ p_x = \pm\sqrt{576} \] \( \sqrt{576} = 24 \) (since \( 24^2 = 576 \)).
So, \( p_x = \pm 24 \) units.
The question asks for "its x-component", implying magnitude, or assumes it's positive.
Options are positive.
So, the x-component is 24 units.
This matches option (1).
If the vector is 3-dimensional, then \( |\vec{p}| = \sqrt{p_x^2 + p_y^2 + p_z^2} \).
\[ 25 = \sqrt{p_x^2 + 7^2 + p_z^2} \] \[ 625 = p_x^2 + 49 + p_z^2 \] \[ p_x^2 + p_z^2 = 625 - 49 = 576 \] In this case, \( p_x \) is not uniquely determined.
For example, if \( p_z = 0 \), then \( p_x^2 = 576 \implies p_x = \pm 24 \).
If \( p_x = 0 \), then \( p_z^2 = 576 \implies p_z = \pm 24 \).
Given typical physics problem contexts where components are asked without specifying dimensionality, it's usually assumed to be the simplest case that fits the given information (2D if only x and y components are mentioned or sought).
The problem does not mention a z-component, so it is common to assume \(p_z=0\).