Two lines $$$l$$$ and $$$m$$$ have equations $$$\mathbf{r}=3 \mathbf{i}+2 \mathbf{j}+5 \mathbf{k}+s(4 \mathbf{i}-\mathbf{j}+3 \mathbf{k})$$$ and $$$\mathbf{r}=\mathbf{i}-\mathbf{j}-2 \mathbf{k}+t(-\mathbf{i}+2 \mathbf{j}+2 \mathbf{k})$$$ respectively. Show that $$$l$$$ and $$$m$$$ intersect and state the position vector of the point of intersection. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mathematics

IGCSE&ALevel

CAIE

Exam No：9709_w21_qp_31 Year：2021 Question No：9(b)

Answer:

Express general point of $l$ or $m$ in component form,
e.g. $(3+4 s, 2-s, 5+3 s)$ or $(1-t,-1+2 t,-2+2 t)$
Equate at least two pairs of components and solve for $s$ or for $t$
Obtain correct answer $s=-1$ and $t=2$
Verify that all three equations are satisfied
State position vector of the intersection $-\mathbf{i}+3 \mathbf{j}+2 \mathbf{k}$, or equivalent

Knowledge points:

3.7.4 understand the significance of all the symbols used when the equation of a straight line is expressed in the form r = a +tb, and find the equation of a line, given sufficient information

3.7.5 determine whether two lines are parallel, intersect or are skew, and find the point of intersection of two lines when it exists (Calculation of the shortest distance between two skew lines is not required. Finding the equation of the common perpendicular to two skew lines is also not required.)

Answer:

Knowledge points:

Solution:

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