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14. A force of [tex]\((3 \vec{i} - 2 \vec{j} + \vec{k}) \, N\)[/tex] produces a displacement of [tex]\((2 \vec{i} - 4 \vec{j} + c \vec{k}) \, m\)[/tex]. The work done is [tex]\(16 \, J\)[/tex]. Then [tex]\(c =\)[/tex]

1) -1

2) -2

3) 1

4) 2



Answer :

GinnyAnswer
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]

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