Midpoint Determination Using Parameter

00:00 5 marks

Consider the vector equation of a line defined by \(\vec{r}(t) = \vec{a} + t(\vec{b} – \vec{a})\), where \(\vec{a}\) and \(\vec{b}\) are position vectors of the endpoints of a line segment.

(a) Express the parameter \(t\) that corresponds to the midpoint of the line segment between \(\vec{a}\) and \(\vec{b}\).

(b) Confirm that when \(t\) is substituted back into \(\vec{r}(t)\), the resulting position vector indeed represents the midpoint.

Part a

3 minutes 2 marks

Question

Express the parameter \(t\) that corresponds to the midpoint of the line segment between \(\vec{a}\) and \(\vec{b}\).

Hint Hint

Hint

Recall that the midpoint is given by the average of the endpoints.

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Complete Answer

\(t = \frac{1}{2}\)

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Part b

5 minutes 3 marks

Question

Confirm that substituting \(t = \frac{1}{2}\) into \(\vec{r}(t)\) yields the midpoint of the line segment.

Hint Hint

Hint

Substitute \(t = \frac{1}{2}\) and simplify the expression.

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Answer Answer

Complete Answer

\(\vec{r}(\tfrac{1}{2}) = \frac{\vec{a} + \vec{b}}{2}\)

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