Question Type 2: Verifying if a point lies on a specific line Exercises

Does the point $P(7, 10, -5)$ lie on the line passing through points $A(1, 2, 3)$ and $B(4, 6, -1)$?

Determine whether the point $(5,3,1)$ lies on the line given by $\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-5}{-2}.$

Determine whether the point $(-1,4,7)$ lies on the parametric line $x=2+3t, \quad y=1+t, \quad z=4+2t$

Determine whether the point $(6,0,3)$ lies on the line $\frac{x-1}{5}=\frac{y+2}{-2}=\frac{z-3}{1}$

Determine whether the point $(4, -1, 2)$ lies on the line $\frac{x}{2} = \frac{y+1}{-3} = \frac{z-5}{4}.$

Determine whether the point $(1, -5, 9)$ lies on the line given by the vector equation $\mathbf{r} = \begin{pmatrix} 3 \\ -2 \\ 1 \end{pmatrix} + t \begin{pmatrix} 2 \\ 1 \\ 4 \end{pmatrix}$

Determine whether $(8,-7,14)$ lies on the line $x=-2+5t, \quad y=3-4t, \quad z=t-1.$

Determine whether the point $(0, -2, 3)$ lies on the vector line defined by $\mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix} + t \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$

Check whether the origin $(0,0,0)$ lies on the line $x=1+t, \quad y=-2+2t, \quad z=3-t.$

Determine whether the point $(10,-4,6)$ lies on the line passing through $(2,5,-1)$ and parallel to the vector $\begin{pmatrix} 4 \\ -3 \\ 2 \end{pmatrix}$.

Determine whether the point $(-8, 0, 10)$ lies on the line given by the equation

[3 marks]

Does the point $(2,5,-1)$ lie on the line through $(0,1,-2)$ and $(4,3,2)$?