IB Mathematics Applications & Interpretation 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 $$x=2+3t,y=1+t,z=4+2t$$

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 $$\mathbf{r}=\left(\begin{array}{c}3\\ -2\\ 1\end{array}\right)+t\left(\begin{array}{c}2\\ 1\\ 4\end{array}\right)$$

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

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

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

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

Determine whether the point $(-8,0,10)$ lies on the line given by the equation $$\frac{x+4}{-2}=\frac{y-2}{1}=\frac{z}{5}.$$ [3 marks]

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