The position vectors of the points A, B and C are vector (2i + j - k),

1.	The position vectors of the points A, B and C are vector (2i + j - k),(3i - 2j + k) and (i+4j-3k) respectively. Show that the points A, B and C are collinear.
Answer» A = \(2\vec{i}+\vec{j}-\vec{k}\) B = \(3\vec{i}-2\vec{j}+\vec{k}\) C = \(\vec{i}+4\vec{j}-3\vec{k}\) \(\vec{AB}\) = \((3\vec{i}-2\vec{j}+\vec{k})\) - \((2\vec{i}+\vec{j}-\vec{k})\) = \(\vec{i}-3\vec{j}+2\vec{k}\) \(\vec{BC}\) \((\vec{i}+4\vec{j}-3\vec{k})\) - \((3\vec{i}-2\vec{j}+\vec{k})\) = \(-2\vec{i}+6\vec{j}-4\vec{k}\) (-3)\(\vec{AB}\) = \(\vec{BC}\) So, the points A, B and C are collinear.

The position vectors of the points A, B and C are vector (2i + j - k),(3i - 2j + k) and (i+4j-3k) respectively. Show that the points A, B and C are collinear.

Answer»

A = \(2\vec{i}+\vec{j}-\vec{k}\)

B = \(3\vec{i}-2\vec{j}+\vec{k}\)

C = \(\vec{i}+4\vec{j}-3\vec{k}\)

\(\vec{AB}\)

= \((3\vec{i}-2\vec{j}+\vec{k})\) - \((2\vec{i}+\vec{j}-\vec{k})\)

= \(\vec{i}-3\vec{j}+2\vec{k}\)

\(\vec{BC}\)

\((\vec{i}+4\vec{j}-3\vec{k})\) - \((3\vec{i}-2\vec{j}+\vec{k})\)

= \(-2\vec{i}+6\vec{j}-4\vec{k}\)

(-3)\(\vec{AB}\) = \(\vec{BC}\)

So, the points A, B and C are collinear.

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