Vector Projection via Partial Coordinates
Consider two vectors defined in the plane: \(\mathbf{v} = (x, 4)\) and \(\mathbf{u} = (2, y)\), where \(x\) and \(y\) are unknown real numbers. It is given that the dot product \(\mathbf{v} \cdot \mathbf{u} = 10\) and that the length of the projection of \(\mathbf{v}\) onto \(\mathbf{u}\) is 3. Given that \(y > 0\), determine […]
Reflecting a Point Across a 3D Line Using Vector Methods
Consider a 3D line L given by the vector equation: \(\vec{r} = \vec{a} + t\vec{d}\), where \(\vec{a} = (1, 2, 3)\) and \(\vec{d} = (2, -1, 2)\). A point P has coordinates (4, 0, 1). (a) Derive the vector equation for the reflection of a general point \(\vec{r}\) across the line L. (b) Using the […]