In introductory kinematics, a particle’s acceleration vector is $\overrightarrow{a}$ with components $a_x$, $a_y$, and $a_z$. How do you find the magnitude of the acceleration?

• A. $\left|\overrightarrow{a}\right|=\frac{dv}{da}$
• B. $\left|\overrightarrow{a}\right|=a_x+a_y+a_z$
• C. $\left|\overrightarrow{a}\right|=\sqrt{a_x+a_y+a_z}$
• D. $\left|\overrightarrow{a}\right|=\sqrt{{a_x}^2+{a_y}^2+{a_z}^2}$

Answer: (D)

Verified step by step guidance

1. Recognize that the acceleration vector \( \vec{a} \) has components along the x, y, and z axes, denoted as \( a_x \), \( a_y \), and \( a_z \) respectively.
2. Recall that the magnitude of a vector in three-dimensional space is found by taking the square root of the sum of the squares of its components.
3. Write the formula for the magnitude of the acceleration vector as: \( |\vec{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2} \)
4. Understand that this formula comes from the Pythagorean theorem extended to three dimensions, which combines the orthogonal components into a single scalar value representing the vector's length.
5. To find the magnitude, square each component \( a_x \), \( a_y \), and \( a_z \), sum these squares, and then take the square root of that sum.