To solve the problem of finding a scalar $t$ such that the scalar multiplication of $t$ with the vector $(0, -2)$ equals $(1, -2)$, let's proceed step by step:

The equation is: $t \cdot (0, -2) = (1, -2)$

This expands to: $(t \cdot 0, t \cdot -2) = (1, -2)$

Step 1: Solve for each component

1. For the $x$-component: $t \cdot 0 = 1 \quad \implies \quad 0 = 1$

   This is a contradiction, meaning there is no scalar $t$ that satisfies the equation.

Conclusion:

It is impossible for any scalar multiplication of $(0, -2)$ to result in $(1, -2)$, as the $x$-component of $(0, -2)$ is always 0 and cannot match the $x$-component of $(1, -2)$.

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Would you like me to elaborate further or address related concepts? Here are 5 questions for deeper exploration:

1. What does scalar multiplication of vectors mean geometrically?
2. Can two vectors be multiplied directly (dot or cross product)?
3. What are the implications when one component of a vector is zero?
4. How do we determine if two vectors are scalar multiples of each other?
5. Can a zero vector be scaled to match any vector?

Tip: When checking if two vectors are scalar multiples, compare all components; any contradiction implies they are not scalar multiples.