To determine if the vectors $\mathbf{a} = -2\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}$ and $\mathbf{b} = 2\mathbf{i} + 4\mathbf{j} - 5\mathbf{k}$ are collinear, we check if one vector is a scalar multiple of the other. That is, we need to see if there exists a scalar $\lambda$ such that:

$\mathbf{b} = \lambda \mathbf{a}$

Given: $\mathbf{a} = -2\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}$ $\mathbf{b} = 2\mathbf{i} + 4\mathbf{j} - 5\mathbf{k}$

We equate the components of $\mathbf{b}$ to $\lambda$ times the components of $\mathbf{a}$:

$2 = \lambda(-2)$ $4 = \lambda(2)$ $-5 = \lambda(-3)$

Solving for $\lambda$ from each equation:

1. $\lambda = \frac{2}{-2} = -1$
2. $\lambda = \frac{4}{2} = 2$
3. $\lambda = \frac{-5}{-3} = \frac{5}{3}$

Since $\lambda$ is not the same for all three components, the vectors are not collinear.

Would you like further details or have any other questions?

Here are five related questions you might be interested in:

1. What is the dot product of two vectors and what does it signify?
2. How can you determine if two vectors are perpendicular?
3. What is the cross product of vectors and how is it used in determining collinearity?
4. How do you find the angle between two vectors?
5. What are the properties of parallel and perpendicular vectors in 3D space?

Tip: When checking for collinearity of vectors, always ensure that the scalar factor between corresponding components is consistent across all dimensions.