#### 36Question: A startup is developing a medical device that uses vector-based sensor arrays to detect physiological signals. Suppose two sensor vectors are given as $\vec{v} = \begin{pmatrix} 3 \\ -2 \\ x \end{pmatrix}$ and $\vec{w} = \begin{pmatrix} x \\ 4 \\ -1 \end{pmatrix}$. Find the value of $x$ such that the vectors are orthogonal.

Sun, 03 May 2026 03:32:58 +0000

Title: How to Determine Orthogonality in 3D Vectors: A Medical Device Sensor Application In the rapidly advancing field of medical device technology, precision in signal detection is critical. One key concept enabling accurate monitoring is the orthogonality of sensor-recorded physiological signalsÃƒÂ¢Ã‚Â€Ã‚Â”ensuring minimal interference and high signal fidelity. A common mathematical challenge in designing such systems involves determining when two sensor vectors are orthogonal, meaning their dot product is zero. Consider two sensor voltage response vectors used in a novel medical monitoring device: $$
ec{v} = egin{pmatrix} 3 \ -2 \ x \end{pmatrix}, \quad ec{w} = egin{pmatrix} x \ 4 \ -1 \end{pmatrix}
$$