To determine the value of [tex]\( n \)[/tex] for which the vector [tex]\(\vec{i} + 2\vec{j} + n\vec{k}\)[/tex] is perpendicular to [tex]\( 4\vec{i} + 2\vec{j} + 2\vec{k} \)[/tex], we need to follow these steps:
1. Recall the condition for two vectors to be perpendicular: their dot product must be zero.
2. Compute the dot product of the given vectors.
3. Set the dot product equal to zero and solve for [tex]\( n \)[/tex].
Let's denote the vectors as:
[tex]\[
\vec{a} = \vec{i} + 2\vec{j} + n\vec{k}
\][/tex]
[tex]\[
\vec{b} = 4\vec{i} + 2\vec{j} + 2\vec{k}
\][/tex]
The dot product of [tex]\(\vec{a}\)[/tex] and [tex]\(\vec{b}\)[/tex] is given by:
[tex]\[
\vec{a} \cdot \vec{b} = (1 \times 4) + (2 \times 2) + (n \times 2)
\][/tex]
Now, let's compute it step-by-step:
[tex]\[
1 \times 4 = 4
\][/tex]
[tex]\[
2 \times 2 = 4
\][/tex]
[tex]\[
n \times 2 = 2n
\][/tex]
Thus, the dot product is:
[tex]\[
\vec{a} \cdot \vec{b} = 4 + 4 + 2n = 8 + 2n
\][/tex]
Since the vectors are perpendicular, their dot product must be zero:
[tex]\[
8 + 2n = 0
\][/tex]
Solving for [tex]\( n \)[/tex]:
[tex]\[
2n = -8
\][/tex]
[tex]\[
n = \frac{-8}{2}
\][/tex]
[tex]\[
n = -4
\][/tex]
Therefore, the value of [tex]\( n \)[/tex] is:
[tex]\[
n = -4
\][/tex]
Thus, the correct answer is:
3) -4
