To determine the value of [tex]\(\lambda\)[/tex] for which the vectors [tex]\(\vec{a} = 3\vec{i} + \vec{j} - \vec{k}\)[/tex] and [tex]\(\vec{b} = \lambda \vec{i} - 4 \vec{j} + 4 \vec{k}\)[/tex] are collinear, we need to satisfy the property of proportionality between the corresponding components of the vectors.
For vectors [tex]\(\vec{a}\)[/tex] and [tex]\(\vec{b}\)[/tex] to be collinear, there must exist a scalar [tex]\(k\)[/tex] such that each component of [tex]\(\vec{a}\)[/tex] is proportional to the corresponding component of [tex]\(\vec{b}\)[/tex]:
[tex]\[
\frac{3}{\lambda} = \frac{1}{-4} = \frac{-1}{4}
\][/tex]
From the equations [tex]\(\frac{3}{\lambda} = \frac{1}{-4}\)[/tex] and [tex]\(\frac{3}{\lambda} = \frac{-1}{4}\)[/tex], we solve for [tex]\(\lambda\)[/tex]:
Step 1: Solve the first proportion:
[tex]\[
\frac{3}{\lambda} = \frac{1}{-4}
\][/tex]
Cross-multiplying gives:
[tex]\[
3 \cdot (-4) = \lambda \cdot 1 \quad \Rightarrow \quad -12 = \lambda
\][/tex]
Step 2: Check the second proportion:
[tex]\[
\frac{3}{\lambda} = \frac{-1}{4}
\][/tex]
Cross-multiplying gives the same result:
[tex]\[
3 \cdot 4 = \lambda \cdot (-1) \quad \Rightarrow \quad 12 = -\lambda \quad \Rightarrow \quad \lambda = -12
\][/tex]
Having confirmed the consistency, we conclude that the value of [tex]\(\lambda\)[/tex] is:
[tex]\[
\lambda = -12
\][/tex]
