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Show that the four points A, B, C and D...

Show that the four points A, B, C and D with position vectors`4 hat i+5\ hat j+ hat k ,-(\ hat j+ hat k),3 hat j+9\ hat j+4 hat k\ a n d\ 4( hat i+ hat j+ hat k)`, respectively are coplanar.

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To show that the four points A, B, C, and D with given position vectors are coplanar, we can use the scalar triple product method. The points are coplanar if the scalar triple product of the vectors formed by these points is zero. ### Step-by-Step Solution: 1. **Identify the Position Vectors:** - Let \( \vec{A} = 4\hat{i} + 5\hat{j} + \hat{k} \) - Let \( \vec{B} = -\hat{j} - \hat{k} \) - Let \( \vec{C} = 3\hat{j} + 9\hat{j} + 4\hat{k} = 12\hat{j} + 4\hat{k} \) (correcting the representation) ...
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