Vector Algebra
Dot Product / Work Done
Grade 12
Question:
<p>Work done = \((\vec{F_1} + \vec{F_2}) \cdot \overrightarrow{AB}\) where \(\overrightarrow{OA} = \hat{i} + 2\hat{j} + 3\hat{k}\), \(\overrightarrow{OB} = 5\hat{i} + 4\hat{j} + \hat{k}\) and \(\vec{F_1} + \vec{F_2} = 7\hat{i} + 2\hat{j} - 4\hat{k}\). Find the work done.</p>
<p>40 units</p>
<p>30 units</p>
<p>50 units</p>
<p>20 units</p>
Step-by-Step Solution
Key Concept: Work done is the dot product of the net force vector with the displacement vector; first find displacement by subtracting position vectors, then compute the scalar product.
Step 1: Find the displacement vector $\overrightarrow{AB}$. $\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA} = (5\hat{i} + 4\hat{j} + \hat{k}) - (\hat{i} + 2\hat{j} + 3\hat{k})$ $\overrightarrow{AB} = 4\hat{i} + 2\hat{j} - 2\hat{k}$ Step 2: Calculate the dot product (work done). $W = (\vec{F_1} + \vec{F_2}) \cdot \overrightarrow{AB} = (7\hat{i} + 2\hat{j} - 4\hat{k}) \cdot (4\hat{i} + 2\hat{j} - 2\hat{k})$ $W = (7)(4) + (2)(2) + (-4)(-2)$ $W = 28 + 4 + 8 = 40$ ∴ Answer: A (Work done = 40 Joules )
Correct Answer: A