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Home » No More Soft Hair? Shampoo Might Be Expired—Here’s How to Spot It Before You Use It! - Carbonext

No More Soft Hair? Shampoo Might Be Expired—Here’s How to Spot It Before You Use It! - Carbonext

📅 February 28, 2026 👤 scraface

Feb 28, 2026

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📌 Solution: In spherical coordinates, $\phi$ is the polar angle measured from the positive $z$-axis. When $\phi$ is constant, the set of points forms a cone with its apex at the origin, opening symmetrically around the $z$-axis. For $\phi = \frac{\pi}{4}$, the cone makes a $45^\circ$ angle with the $z$-axis. Thus, the shape is a cone.

📌 \boxed{\text{Cone}}

📌 Question: A synthetic biology researcher models microbial growth directions in permafrost using unit vectors $\mathbf{u}, \mathbf{v}, \mathbf{w}$. If $\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{w} = \mathbf{w} \cdot \mathbf{u} = \frac{1}{2}$, find the maximum value of $|\mathbf{u} + \mathbf{v} + \mathbf{w}|$.

📌 Solution: Let $\theta$ be the angle between any pair of vectors. Since $\mathbf{u} \cdot \mathbf{v} = \cos\theta = \frac{1}{2}$, $\theta = 60^\circ$. The magnitude squared is $|\mathbf{u} + \mathbf{v} + \mathbf{w}|^2 = 3 + 2(\mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} + \mathbf{v} \cdot \mathbf{w}) = 3 + 2(3 \cdot \frac{1}{2}) = 3 + 3 = 6$. Thus, the maximum value is $\sqrt{6}$.

📌 Question: A plant biologist studies stress responses using vectors $\begin{pmatrix} 1 \\ x \\ 2 \end{pmatrix}$ and $\begin{pmatrix} x \\ 3 \\ -1 \end{pmatrix}$. Find $x$ such that the vectors are orthogonal.

📌 Solution: For orthogonality, their dot product must be zero: $1 \cdot x + x \cdot 3 + 2 \cdot (-1) = 0$. Simplifying, $x + 3x - 2 = 0 \Rightarrow 4x = 2 \Rightarrow x = \frac{1}{2}$.

📌 \boxed{\dfrac{1}{2}}

📌 Question: In a synthetic biology diagram, $\|\overrightarrow{OA}\| = 2$, $\|\overrightarrow{OB}\| = 3$, and $\angle AOB = 90^\circ$. If $\overrightarrow{OC} = 2\overrightarrow{OA} - \overrightarrow{OB}$, find $\|\overrightarrow{OC}\|$.

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