Vector Algebra
Moment and Cross Product
Grade 12
Question:
<p>The moment about the point <span>\(\vec{i} + 2\vec{j} + 3\vec{k}\)</span> of a force represented by <span>\(\vec{i} + \vec{j} + \vec{k}\)</span> acting through the point <span>\(2\vec{i} + 3\vec{j} + \vec{k}\)</span>, is</p>
<p>(a) <span>\(3\vec{i} + 3\vec{j}\)</span></p>
<p>(b) <span>\(3\vec{i} + \vec{j}\)</span></p>
<p>(c) <span>\(\vec{i} - \vec{j}\)</span></p>
<p>(d) <span>\(3\vec{i} - 3\vec{j}\)</span></p>
Step-by-Step Solution
Key Concept: The moment of a force about a point is calculated using the cross product of the position vector from the point of application and the force vector.
Step 1: The position vector from the point of application to the point about which moment is calculated is: $\vec{r} = (2\hat{i} + 3\hat{j} + \hat{k}) - (\hat{i} + 2\hat{j} + 3\hat{k})$ $\vec{r} = \hat{i} + \hat{j} - 2\hat{k}$ Step 2: The force is $\vec{F} = \hat{i} + \hat{j} + \hat{k}$ Step 3: The moment is given by $\vec{r} \times \vec{F}$ : $\vec{r} \times \vec{F} = (\hat{i} + \hat{j} - 2\hat{k}) \times (\hat{i} + \hat{j} + \hat{k})$ $= \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & -2 \\ 1 & 1 & 1 \end{vmatrix}$ $= 3\hat{i} - 3\hat{j}$ ∴ Answer is (d) $3\hat{i} - 3\hat{j}$ .
Correct Answer: D