Midpoint Determination Using Parameter
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 […]
Part a
Express the parameter \(t\) that corresponds to the midpoint of the line segment between \(\vec{a}\) and \(\vec{b}\).
Part b
Confirm that substituting \(t = \frac{1}{2}\) into \(\vec{r}(t)\) yields the midpoint of the line segment.
Expressing Vectors: Component and Parametric Representations
Given the vector \(\mathbf{v} = (6, -3)\), express it in component form and write the corresponding parametric equations for the line passing through the origin in the direction of \(\mathbf{v}\).