6. A particle undergoes three consecutive displacements: $$ \bar{d}_{1}=(15 i+30 j+12 k) \mathrm{cm} $$. $$ \vec{d}_{2}=(23 i-14 j-5 k) a n $$ and $$ \vec{d}_{3}=(-13 i+15 j) $$ an Find the magnitude of the resultant displacement.

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UpStudy AI · Accepted Answer

To find the magnitude of the resultant displacement, we'll follow these steps:

1. **Identify the Displacement Vectors:**
   - $$\bar{d}_1 = 15\mathbf{i} + 30\mathbf{j} + 12\mathbf{k}$$ cm
   - $$\bar{d}_2 = 23\mathbf{i} - 14\mathbf{j} - 5\mathbf{k}$$ cm
   - $$\bar{d}_3 = -13\mathbf{i} + 15\mathbf{j}$$ cm

2. **Sum the Displacement Vectors:**
   $$
   \bar{R} = \bar{d}_1 + \bar{d}_2 + \bar{d}_3
   $$
   $$
   \bar{R} = (15 + 23 - 13)\mathbf{i} + (30 - 14 + 15)\mathbf{j} + (12 - 5 + 0)\mathbf{k}
   $$
   $$
   \bar{R} = 25\mathbf{i} + 31\mathbf{j} + 7\mathbf{k}
   $$

3. **Calculate the Magnitude of the Resultant Displacement:**
   $$
   |\bar{R}| = \sqrt{(25)^2 + (31)^2 + (7)^2}
   $$
   $$
   |\bar{R}| = \sqrt{625 + 961 + 49}
   $$
   $$
   |\bar{R}| = \sqrt{1635}
   $$
   $$
   |\bar{R}| \approx 40.43 \text{ cm}
   $$

**Final Answer:**
The magnitude of the resultant displacement is approximately 40.43 cm.
The resultant displacement is about 40.43 cm.

A particle undergoes three consecutive displacements: .
and an Find the magnitude of the resultant
displacement.

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A particle undergoes three consecutive displacements: . and an Find the magnitude of the resultant displacement.