Question Type 1: Finding vector equation of a line passing through two points Exercises

Determine if point $N(7, 4, 1)$ lies on the line defined by the equation

$\frac{x+1}{4} = \frac{y-2}{-1} = \frac{z-3}{2}$

Determine whether the point $(4,-2,5)$ lies on the line $\vec{r}=\begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}+t\begin{pmatrix} 3 \\ -1 \\ 1 \end{pmatrix}\,.$

Verify whether the point $M(2,1,3)$ lies on the line given by $\frac{x-5}{2}=\frac{y-1}{3}=\frac{z}{1}$

Express the line $\vec{r} = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix} + s\begin{pmatrix} -5 \\ 4 \\ 2 \end{pmatrix}$ in symmetric form.

Find the vector equation of the line passing through the points $P(1,4,3)$ and $Q(-1,2,1)$.

Find the parametric equations of the line $\vec{r} = \begin{pmatrix} 0 \\ 1 \\ 6 \end{pmatrix} + k \begin{pmatrix} 4 \\ 2 \\ 1 \end{pmatrix}$.

Convert the line $\frac{x-3}{1} = \frac{y+2}{-2} = \frac{z-5}{4}$ into vector form.

Find the vector equation of the line joining $E(3, 0, 1)$ to $F(-1, 4, -2)$.

Verify whether $(0, 3, 4)$ lies on the line through $(2, 1, 2)$ in direction $(-2, 2, 3)$.

Find the parametric equations of the line with Cartesian equation $\frac{x}{2}=\frac{y-3}{3}=\frac{z+1}{-1}$

Write the vector equation of the line through $C(0,5,-2)$ and $D(-4,1,-6)$.

Determine a vector equation of the line through points $A(2, -1, 0)$ and $B(5, 2, 3)$.