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Transcribed Image Text:**Educational Content for HW 14 on Vectors**
**Instructions:**
Use the information about vectors \( \vec{S} \) and \( \vec{T} \) provided below in the following exercises.
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**Vector Information:**
- \( |\vec{S}| = 12.8 \, m \)
- \( \vec{S} \cdot \hat{x} = 2 \, m \)
- \( \vec{S} \cdot \hat{y} = 10 \, m \)
- \( \vec{S} \cdot \hat{z} = -8 \, m \)
- \( |\vec{T}| = 10 \, m \)
- \( \vec{T} \cdot \hat{x} = 0 \, m \)
- \( \vec{T} \cdot \hat{y} = -3 \, m \)
- \( \vec{T} \cdot \hat{z} = 0 \, m \)
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**Chosen Coordinate System Diagram:**
- The coordinate system shows three unit vectors:
- \( \hat{x} \) is pointing upwards.
- \( \hat{y} \) is represented by a circle with a dot in the center (indicating out of the page).
- \( \hat{z} \) is pointing to the right.
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**Exercises:**
a) *Draw the direction of the missing Cartesian unit vector \( \hat{z} \).*
b) *Write the expression for the vector component of \( \vec{S} \) in the \( \hat{z} \) direction.*
c) *Determine the smallest angle between \( \vec{S} \) and \( \hat{z} \), when they are drawn tail to tail.*
d) *Express \( \vec{S} \) and \( \vec{T} \) in component form, using the chosen coordinate system.*
e) *Determine \( \vec{S} \cdot \vec{T} \) (the dot product).*
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