Part a
Extract the direction vectors from the given parametric equations of the lines: Line 1: \(\mathbf{r} = (1, 2, 3) + t(1, -1, 2)\) Line 2: \(\mathbf{r} = (4, 0, -1) + s(2, 3, -1)\)
Calculating the Angle Between Two Lines Using Dot Product
Consider the following two lines in space: Line 1: \(\mathbf{r} = (1, 2, 3) + t(1, -1, 2)\) Line 2: \(\mathbf{r} = (4, 0, -1) + s(2, 3, -1)\) Perform the following tasks: (a) Extract the direction vectors for each line from their parametric equations. (b) Using the dot product, calculate the angle between the […]
Part b
Explain why any non-zero scalar multiple of the direction vector obtained in part (a) would also be a valid direction vector for the same line. Justify your answer.
Optimizing and Relating Vector Equations in Plane Geometry
Consider a plane defined by a point P and a normal vector \(\textbf{n}\). A line, given by its parametric equation, intersects this plane. Given a fixed point Q, determine the position of the plane that minimizes the distance from Q to the plane. Subsequently, analyze the relationship between the direction vector of the intersecting line […]
Part a
Determine the relationships among the three lines given below: L1: \(x = 2 + t, \; y = 1 – t, \; z = 5 + 2t\). L2: \(x = 2 + 3s, \; y = 1 + 2s, \; z = 5 – s\). L3: \(x = r, \; y = r, \; z […]
Determining Skew Lines in Space
Given two lines in 3-dimensional space: Line L1: \(\mathbf{r}_1 = \langle 2, -1, 3 \rangle + t\langle 1, 2, -1 \rangle\) Line L2: \(\mathbf{r}_2 = \langle -1, 4, 0 \rangle + s\langle 2, -1, 1 \rangle\) Determine whether these two lines are skew. Proceed with the following steps: (a) Compare the direction vectors of both […]
Parallel Line Identification in 3D
Given two lines in 3D: Line L1: passes through the point (1, 2, 3) with a direction vector \(\begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix}\). Line L2: passes through the point (4, 5, 6) with a direction vector \(\begin{pmatrix} 2 \\ -4 \\ 6 \end{pmatrix}\). Determine whether these two lines are parallel.
Vector Equation of a Line in Motion
A car is moving along a straight road. At time \(t = 0\), its position is given by the point \(P(2,3)\). The direction of motion is along the vector \(\langle 3,4 \rangle\). Write the vector equation of the line representing the car’s trajectory.
Scaling and Simplifying Vector Equations
Given the vector equation of a line \(\mathbf{r} = \mathbf{a} + t\mathbf{d}\), where \(\mathbf{a} = (2, -3)\) and \(\mathbf{d} = (6, -9)\), rewrite the equation by scaling the direction vector to its simplest form.
Checking Parallelism with Direction Vectors
Given the line \(r: \vec{x} = \begin{pmatrix} 2 \\ -3 \end{pmatrix} + t\begin{pmatrix} 4 \\ 5 \end{pmatrix}\) and the line \(s: \vec{x} = \begin{pmatrix} -1 \\ 1 \end{pmatrix} + u\begin{pmatrix} -8 \\ -10 \end{pmatrix}\), determine if the two lines are parallel. Justify your answer by comparing their direction vectors.