To find the value of [tex]\( c \)[/tex] in the displacement vector [tex]\((2 \vec{i} - 4 \vec{j} + c \vec{k})\)[/tex] given that the work done by the force [tex]\((3 \vec{i} - 2 \vec{j} + \vec{k})\)[/tex] is 16 Joules, we can use the formula for work done by a force:
[tex]\[ W = \vec{F} \cdot \vec{d} \][/tex]
where [tex]\( \vec{F} \)[/tex] is the force vector and [tex]\( \vec{d} \)[/tex] is the displacement vector. The dot product of two vectors [tex]\(\vec{F}\)[/tex] and [tex]\(\vec{d}\)[/tex] is calculated as:
[tex]\[ \vec{F} \cdot \vec{d} = F_x \cdot d_x + F_y \cdot d_y + F_z \cdot d_z \][/tex]
Given:
[tex]\[ \vec{F} = (3 \vec{i} - 2 \vec{j} + \vec{k}) \][/tex]
[tex]\[ \vec{d} = (2 \vec{i} - 4 \vec{j} + c \vec{k}) \][/tex]
[tex]\[ W = 16 \text{ J} \][/tex]
We can substitute these vectors into the dot product formula:
[tex]\[ \vec{F} \cdot \vec{d} = (3 \cdot 2) + (-2 \cdot -4) + (1 \cdot c) \][/tex]
Calculating step by step:
1. The [tex]\( \vec{i} \)[/tex] components:
[tex]\[ 3 \cdot 2 = 6 \][/tex]
2. The [tex]\( \vec{j} \)[/tex] components:
[tex]\[ -2 \cdot -4 = 8 \][/tex]
3. The [tex]\( \vec{k} \)[/tex] components:
[tex]\[ 1 \cdot c = c \][/tex]
Combining these results:
[tex]\[ 6 + 8 + c = 16 \][/tex]
Simplify the equation:
[tex]\[ 14 + c = 16 \][/tex]
To find [tex]\( c \)[/tex]:
[tex]\[ c = 16 - 14 \][/tex]
[tex]\[ c = 2 \][/tex]
Therefore, the value of [tex]\( c \)[/tex] is:
[tex]\[ \boxed{2} \][/tex]
