IB Mathematics Analysis and Approaches Question Type 1: Finding the Point of Intersection for Two Lines Exercises

Find the coordinates of the point of intersection of the lines $L_1$ and $L_2$ defined by the equations: $$L_1:\mathbf{r}=\left(\begin{array}{c}-2\\ 1\\ 3\end{array}\right)+t\left(\begin{array}{c}4\\ -3\\ 2\end{array}\right)$$ $$L_2:\mathbf{r}=\left(\begin{array}{c}4\\ -7\\ 8\end{array}\right)+u\left(\begin{array}{c}-2\\ 5\\ -3\end{array}\right)$$

Find the point of intersection of the lines $L_1$ and $L_2$ defined by:

$$L_1:\mathbf{r}=\left(\begin{array}{c}1\\ 0\\ 2\end{array}\right)+t\left(\begin{array}{c}2\\ 1\\ -1\end{array}\right)$$

$$L_2:\mathbf{r}=\left(\begin{array}{c}1\\ 1\\ -1\end{array}\right)+u\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right)$$

Determine the coordinates of the intersection of the lines $L_1$ and $L_2$ defined by:

$$L_1:\mathbf{r}=\left(\begin{array}{c}2\\ 3\\ 0\end{array}\right)+t\left(\begin{array}{c}3\\ -2\\ 1\end{array}\right)$$

$$L_2:\mathbf{r}=\left(\begin{array}{c}12\\ -7\\ 1\end{array}\right)+u\left(\begin{array}{c}-1\\ 4\\ 2\end{array}\right)$$

Find the coordinates of the point of intersection of the lines $L_1$ and $L_2$ defined by the following equations:

$$L_1:\mathbf{r}=\left(\begin{array}{c}0\\ 1\\ 3\end{array}\right)+t\left(\begin{array}{c}1\\ 2\\ -1\end{array}\right)$$

$$L_2:\mathbf{r}=\left(\begin{array}{c}4\\ 4\\ 2\end{array}\right)+u\left(\begin{array}{c}2\\ -1\\ 1\end{array}\right)$$

Determine the coordinates of the intersection of the lines $L_1$ and $L_2$: $$L_1:\mathbf{r}=(1,1,1)+t(5,2,-1)$$ $$L_2:\mathbf{r}=(2,17,-7)+u(3,-4,2)$$

Determine the coordinates of the point of intersection of the following two lines:

$$L_1:\mathbf{r}=\left(\begin{array}{c}2\\ 1\\ 0\end{array}\right)+t\left(\begin{array}{c}3\\ -1\\ 2\end{array}\right)$$

$$L_2:\mathbf{r}=\left(\begin{array}{c}5\\ -2\\ 4.5\end{array}\right)+u\left(\begin{array}{c}1\\ 1\\ -1\end{array}\right)$$

Find the coordinates of the intersection point.

Find the coordinates of the point of intersection of the two lines.

Find the intersection point of the lines: $$L_1:\mathbf{r}=\left(\begin{array}{c}0\\ 0\\ 0\end{array}\right)+t\left(\begin{array}{c}1\\ 2\\ 3\end{array}\right),L_2:\mathbf{r}=\left(\begin{array}{c}3\\ 4\\ 5\end{array}\right)+u\left(\begin{array}{c}-2\\ -3\\ -4\end{array}\right)$$

Find the coordinates of the point of intersection of the lines $L_1$ and $L_2$.

$$L_1:\mathbf{r}=\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)+t\left(\begin{array}{c}1\\ 2\\ 3\end{array}\right)$$

$$L_2:\mathbf{r}=\left(\begin{array}{c}2.5\\ 3.5\\ 6\end{array}\right)+u\left(\begin{array}{c}2\\ 1\\ 1\end{array}\right)$$

Calculate the intersection point of $$L_1:(-1,0,4)+t(2,1,3),L_2:(-1,-1,9)+u(1,1,-1).$$